Sample records for explicit runge-kutta method

  1. Composite group of explicit Runge-Kutta methods

    Hamid, Fatin Nadiah Abd; Rabiei, Faranak; Ismail, Fudziah


    In this paper,the composite groups of Runge-Kutta (RK) method are proposed. The composite group of RK method of third and second order, RK3(2) and fourth and third order RK4(3) base on classical Runge-Kutta method are derived. The proposed methods are two-step in nature and have less number of function evaluations compared to the existing Runge-Kutta method. The order conditions up to order four are obtained using rooted trees and composite rule introduced by J. C Butcher. The stability regions of RK3(2) and RK4(3) methods are presented and initial value problems of first order ordinary differential equations are carried out. Numerical results are compared with existing Runge-Kutta method.

  2. Exponentially fitted explicit Runge-Kutta-Nystrom methods

    Franco, J. M.


    Exponentially fitted Runge-Kutta-Nystrom (EFRKN) methods for the numerical integration of second-order IVPs with oscillatory solutions are derived. These methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(λt),exp(-λt)}, , or equivalently {sin(ωt),cos(ωt)} when λ=iω, . Explicit EFRKN methods with two and three stages and algebraic orders 3 and 4 are constructed. In addition, a 4(3) embedded pair of explicit EFRKN methods based on the FSAL technique is obtained, which permits to introduce an error and step length control with a small cost added. Some numerical experiments show the efficiency of our explicit EFRKN methods when they are compared with other exponential fitting type codes proposed in the scientific literature.

  3. Explicit Singly Diagonally Implicit Runge-Kutta Methods and Adaptive Stepsize Control for Reservoir Simulation

    Völcker, Carsten; Jørgensen, John Bagterp; Thomsen, Per Grove


    control applied to high order methods for temporal discretization in reservoir simulation. The family of Runge-Kutta methods is presented and in particular the explicit singly diagonally implicit Runge-Kutta (ESDIRK) method with an embedded error estimate is described. A predictive stepsize adjustment...... rule based on error estimates and convergence control of the integrated iterative solver is presented. We try to improve the predictive stepsize control through an extended communication between the convergence rate, the error control and the stepsize. Keywords: Reservoir simulation, implicit Runge-Kutta...

  4. Internal Error Propagation in Explicit Runge--Kutta Methods

    Ketcheson, David I.


    In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.

  5. Strong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order

    Hadjimichael, Yiannis


    We apply the concept of effective order to strong stability preserving (SSP) explicit Runge--Kutta methods. Relative to classical Runge--Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods---like classical order five methods---require the use of nonpositive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge--Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.

  6. Second-order stabilized explicit Runge-Kutta methods for stiff problems

    Martín-Vaquero, J.; Janssen, B.


    Stabilized Runge-Kutta methods (they have also been called Chebyshev-Runge-Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge-Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10 5). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.

  7. Explicit Singly Diagonally Implicit Runge-Kutta Methods and Adaptive Stepsize Control for Reservoir Simulation

    Völcker, Carsten; Jørgensen, John Bagterp; Thomsen, Per Grove


    The implicit Euler method, normally refered to as the fully implicit (FIM) method, and the implicit pressure explicit saturation (IMPES) method are the traditional choices for temporal discretization in reservoir simulation. The FIM method offers unconditionally stability in the sense of discrete....... Current reservoir simulators apply timestepping algorithms that are based on safeguarded heuristics, and can neither guarantee convergence in the underlying equation solver, nor provide estimates of the relations between convergence, integration error and stepsizes. We establish predictive stepsize...... control applied to high order methods for temporal discretization in reservoir simulation. The family of Runge-Kutta methods is presented and in particular the explicit singly diagonally implicit Runge-Kutta (ESDIRK) method with an embedded error estimate is described. A predictive stepsize adjustment...

  8. Optimized low-order explicit Runge-Kutta schemes for high- order spectral difference method

    Parsani, Matteo


    Optimal explicit Runge-Kutta (ERK) schemes with large stable step sizes are developed for method-of-lines discretizations based on the spectral difference (SD) spatial discretization on quadrilateral grids. These methods involve many stages and provide the optimal linearly stable time step for a prescribed SD spectrum and the minimum leading truncation error coefficient, while admitting a low-storage implementation. Using a large number of stages, the new ERK schemes lead to efficiency improvements larger than 60% over standard ERK schemes for 4th- and 5th-order spatial discretization.

  9. Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems

    Parsani, M; Deconinck, W


    Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge-Kutta schemes available in literature. Furthermore, they have a small principal error norm and admit a low-storage implementation. The advantages of the new schemes are demonstrated through application to the Euler equations and the linearized Euler equations.

  10. Optimized Explicit Runge--Kutta Schemes for the Spectral Difference Method Applied to Wave Propagation Problems

    Parsani, Matteo


    Explicit Runge--Kutta schemes with large stable step sizes are developed for integration of high-order spectral difference spatial discretizations on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge--Kutta schemes available in the literature. Furthermore, they have a small principal error norm and admit a low-storage implementation. The advantages of the new schemes are demonstrated through application to the Euler equations and the linearized Euler equations.

  11. Simulation of 2-D Compressible Flows on a Moving Curvilinear Mesh with an Implicit-Explicit Runge-Kutta Method

    AbuAlSaud, Moataz


    The purpose of this thesis is to solve unsteady two-dimensional compressible Navier-Stokes equations for a moving mesh using implicit explicit (IMEX) Runge- Kutta scheme. The moving mesh is implemented in the equations using Arbitrary Lagrangian Eulerian (ALE) formulation. The inviscid part of the equation is explicitly solved using second-order Godunov method, whereas the viscous part is calculated implicitly. We simulate subsonic compressible flow over static NACA-0012 airfoil at different angle of attacks. Finally, the moving mesh is examined via oscillating the airfoil between angle of attack = 0 and = 20 harmonically. It is observed that the numerical solution matches the experimental and numerical results in the literature to within 20%.


    Geng Sun


    With the help of symplecticity conditions of Partitioned Runge-Kutta methods, a simple way constructing symplectic methods is derived. Examples including sev eral classes of high order symplectic Runge-Kutta methods are given, and showed up the relationship between existing high order Runge-Kutta methods.

  13. Stability of Runge-Kutta-Nystrom methods

    Alonso-Mallo, I.; Cano, B.; Moreta, M. J.


    In this paper, a general and detailed study of linear stability of Runge-Kutta-Nystrom (RKN) methods is given. In the case that arbitrarily stiff problems are integrated, we establish a condition that RKN methods must satisfy so that a uniform bound for stability can be achieved. This condition is not satisfied by any method in the literature. Therefore, a stable method is constructed and some numerical comparisons are made.

  14. Trigonometrical fitting conditions for two derivative Runge Kutta methods

    Monovasilis, Th.; Kalogiratou, Z.; Simos, T. E.


    Trigonometrically fitted two derivative explicit Runge-Kutta methods are considered in this work. We give order conditions for trigonometrically fitted methods that use several evaluations of the f and the g functions. We present modified methods based on methods with several f evaluations and one g evaluation.

  15. Order reduction and how to avoid it when explicit Runge-Kutta-Nystrom methods are used to solve linear partial differential equations

    Alonso-Mallo, I.; Cano, B.; Moreta, M. J.


    In this paper, we study the order reduction which turns up when explicit Runge-Kutta-Nystrom methods are used to discretize linear second order hyperbolic equations by means of the method of lines. The order observed in practice, including its fractional part, is obtained. It is also proved that the order reduction can be completely avoided taking the boundary values of the intermediate stages of the time semidiscretization. The numerical experiments confirm that the optimal order can be reached.

  16. Runge-Kutta collocation methods for differential-algebraic equations of indices 2 and 3

    Skvortsov, L. M.


    Stiffly accurate Runge-Kutta collocation methods with explicit first stage are examined. The parameters of these methods are chosen so as to minimize the errors in the solutions to differential-algebraic equations of indices 2 and 3. This construction results in methods for solving such equations that are superior to the available Runge-Kutta methods.

  17. Spatially Partitioned Embedded Runge--Kutta Methods

    Ketcheson, David I.


    We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in nonembedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to nonphysical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted nonoscillatory spatial discretizations. Numerical experiments are provided to support the theory.

  18. Some procedures for the construction of high-order exponentially fitted Runge-Kutta-Nyström methods of explicit type

    Franco, J. M.; Gómez, I.


    The construction of high-order exponentially fitted Runge-Kutta-Nyström (EFRKN) methods of explicit type for the numerical solution of oscillatory differential systems is analyzed. Based on two basic symmetric and symplectic EFRKN methods of reference we present two procedures for constructing high-order explicit methods. The first procedure is based on composition methods and it allows the construction of high-order explicit EFRKN methods which are symmetric and symplectic. The second procedure is based on combining different EFRKN methods in order to construct embedded pairs of explicit parallel EFRKN methods which can be implemented in variable-step codes without additional cost. The numerical experiments carried out show the qualitative behavior and the efficiency of the new EFRKN methods when they are compared with some standard methods proposed in the scientific literature for solving second-order nonstiff differential systems. Catalogue identifier: AEOO_v1_0 Program summary URL: Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 2527 No. of bytes in distributed program, including test data, etc.: 107433 Distribution format: tar.gz Programming language: Fortran 77. Computer: Standard PC. Operating system: Windows. It might work with others. Successfully tested by CPC on Linux. RAM: For the test problems used less than 1 MB. Classification: 4.3, 4.12, 16.3, 17.17. Nature of problem: Some models in astronomy and astrophysics, quantum mechanics and nuclear physics lead to second-order oscillatory differential systems. The solution of these oscillatory models requires accurate and efficient numerical methods. The codes SVI-IIEXPOreferee.for and SVI-IIvarreferee.for were developed for this purpose. Solution method: We propose high-order exponentially fitted Runge-Kutta

  19. A Generalized Runge-Kutta Method of order three

    Thomsen, Per Grove


    The report presents a numerical method for the solution of stiff systems of ODE's and index one DAE's. The type of method is a 4- stage Generalized Linear Method that is reformulated in a special Semi Implicit Runge Kutta Method of SDIRK type. Error estimation is by imbedding a method of order 4...

  20. Runge-Kutta methods and viscous wave equations

    J.G. Verwer (Jan)


    htmlabstractWe study the numerical time integration of a class of viscous wave equations by means of Runge-Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection-diffusion terms differentiated in time. The viscous wave equation can be

  1. Partially implicit Runge-Kutta methods for wave-like equations in spherical-type coordinates

    Cordero-Carrión, Isabel


    Partially implicit Runge-Kutta methods are presented in this work in order to numerically evolve in time a set of partial differential equations. These methods are designed to overcome numerical instabilities appearing during the evolution of a system of equations due to potential numerical unstable terms in the sources, such as stiff terms or the presence of factors as a result of a particular chosen system of coordinates. In this article, partially implicit Runge-Kutta methods for several convergence orders have been derived and stability properties have been analyzed. These methods are shown to be appropriated to avoid the development of numerical instabilities in the evolution in time of wave-like equations in spherical-type coordinates, in contrast to the explicit Runge-Kutta methods.

  2. Explicit Runge-Kutta schemes for incompressible flow with improved energy-conservation properties

    Capuano, F.; Coppola, G.; Rández, L.; de Luca, L.


    The application of pseudo-symplectic Runge-Kutta methods to the incompressible Navier-Stokes equations is discussed in this work. In contrast to fully energy-conserving, implicit methods, these are explicit schemes of order p that preserve kinetic energy to order q, with q > p. Use of explicit methods with improved energy-conservation properties is appealing for convection-dominated problems, especially in case of direct and large-eddy simulation of turbulent flows. A number of pseudo-symplectic methods are constructed for application to the incompressible Navier-Stokes equations and compared in terms of accuracy and efficiency by means of numerical simulations.

  3. Agglomeration multigrid methods with implicit Runge-Kutta smoothers applied to aerodynamic simulations on unstructured grids

    Langer, Stefan


    For unstructured finite volume methods an agglomeration multigrid with an implicit multistage Runge-Kutta method as a smoother is developed for solving the compressible Reynolds averaged Navier-Stokes (RANS) equations. The implicit Runge-Kutta method is interpreted as a preconditioned explicit Runge-Kutta method. The construction of the preconditioner is based on an approximate derivative. The linear systems are solved approximately with a symmetric Gauss-Seidel method. To significantly improve this solution method grid anisotropy is treated within the Gauss-Seidel iteration in such a way that the strong couplings in the linear system are resolved by tridiagonal systems constructed along these directions of strong coupling. The agglomeration strategy is adapted to this procedure by taking into account exactly these anisotropies in such a way that a directional coarsening is applied along these directions of strong coupling. Turbulence effects are included by a Spalart-Allmaras model, and the additional transport-type equation is approximately solved in a loosely coupled manner with the same method. For two-dimensional and three-dimensional numerical examples and a variety of differently generated meshes we show the wide range of applicability of the solution method. Finally, we exploit the GMRES method to determine approximate spectral information of the linearized RANS equations. This approximate spectral information is used to discuss and compare characteristics of multistage Runge-Kutta methods.

  4. Dissipativity of Multistep Runge-Kutta Methods for Nonlinear Volterra Delay-integro-differential Equations

    Rui QI; Cheng-jian ZHANG; Yu-jie ZHANG


    This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. We investigate the dissipativity properties of (k,l)-algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid.The finitedimensional and infinite-dimensional dissipativity results of (k,l)-algebraically stable Runge-Kutta methods are obtained.


    Xiao-hua Ding; Mingzhu Liu


    Implicit Runge-Kutta method is highly accurate and stable for stiff initial value prob-lem. But the iteration technique used to solve implicit Runge-Kutta method requires lots of computational efforts. In this paper, we extend the Parallel Diagonal Iterated Runge-Kutta(PDIRK) methods to delay differential equations(DDEs). We give the convergence region of PDIRK methods, and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector method. Finally, we analysis the speed-up factor through a numerical experiment. The results show that the PDIRK methods to DDEs are efficient.

  6. Symmetric Uniformly Accurate Gauss-Runge-Kutta Method

    Dauda G. YAKUBU


    Full Text Available Symmetric methods are particularly attractive for solving stiff ordinary differential equations. In this paper by the selection of Gauss-points for both interpolation and collocation, we derive high order symmetric single-step Gauss-Runge-Kutta collocation method for accurate solution of ordinary differential equations. The resulting symmetric method with continuous coefficients is evaluated for the proposed block method for accurate solution of ordinary differential equations. More interestingly, the block method is self-starting with adequate absolute stability interval that is capable of producing simultaneously dense approximation to the solution of ordinary differential equations at a block of points. The use of this method leads to a maximal gain in efficiency as well as in minimal function evaluation per step.

  7. Runge-Kutta methods with minimum storage implementations

    Ketcheson, David I.


    Solution of partial differential equations by the method of lines requires the integration of large numbers of ordinary differential equations (ODEs). In such computations, storage requirements are typically one of the main considerations, especially if a high order ODE solver is required. We investigate Runge-Kutta methods that require only two storage locations per ODE. Existing methods of this type require additional memory if an error estimate or the ability to restart a step is required. We present a new, more general class of methods that provide error estimates and/or the ability to restart a step while still employing the minimum possible number of memory registers. Examples of such methods are found to have good properties. © 2009 Elsevier Inc. All rights reserved.


    Cheng-ming Huang; Hong-yuan Fu; Shou-fu Li; Guang-nan Chen


    This paper is concerned with the numerical solution of delay differential equations(DDEs).We focus on the error behaviour of Runge-Kutta methods for stiff DDEs. We investigate D-convergence properties of algebraically stable Runge-Kutta methods with three kinds of interpolation procedures.

  9. On the multisymplecticity of partitioned Runge-Kutta and splitting methods

    B.N. Ryland; R.I. McLachlan; J.E. Frank (Jason)


    htmlabstractAlthough Runge-Kutta and partitioned Runge-Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrödinger

  10. Diagonally implicit Runge-Kutta methods for 3D shallow water applications

    P.J. van der Houwen; B.P. Sommeijer (Ben)


    textabstractWe construct A-stable and L-stable diagonally implicit Runge-Kutta methods of which the diagonal vector in the Butcher matrix has a minimal maximum norm. If the implicit Runge-Kutta relations are iteratively solved by means of the approximately factorized Newton process, then such


    Yue-xin Yu; Shou-fu Li


    This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on (k, l)-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.

  12. Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations


    The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.

  13. Stability Analysis of Runge-Kutta Methods for Delay Integro-Differential Equations

    甘四清; 郑纬民


    Considering a linear system of delay integro-differential equations with a constant delay whose zero solution is asympototically stable, this paper discusses the stability of numerical methods for the system. The adaptation of Runge-Kutta methods with a Lagrange interpolation procedure was focused on inheriting the asymptotic stability of underlying linear systems. The results show that an A-stable Runge-Kutta method preserves the asympototic stability of underlying linear systems whenever an unconstrained grid is used.

  14. B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations

    LI; Shoufu(李寿佛)


    B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differentialequations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs ofother type which appear in practice.

  15. Semi-implicit Runge.Kutta Method for Solving Stiff Ordinary Differential Equations

    LONGYongxing; MOUZongze; DONGJiaqi; ZHAOHuaiguo


    Runge-Kutta method is widely applied to solve the initial value problem of ordinary differential equations. The implicitRunge-Kutta with better numerical stability for the numerical integration of stiff differential systems,but the formulate has traditionally been on solving the nonlinear equations resulting from a modified Newton iteration in every time.Semi-implicit formulate have the major computationally advantage that it is necessary to solve only linear systems of algebraic equations to find the Ka.

  16. Finite Element Modeling of Thermo Creep Processes Using Runge-Kutta Method

    Yu. I. Dimitrienko


    Full Text Available Thermo creep deformations for most heat-resistant alloys, as a rule, nonlinearly depend on stresses and are practically non- reversible. Therefore, to calculate the properties of these materials the theory of plastic flow is most widely used. Finite-element computations of a stress-strain state of structures with account of thermo creep deformations up to now are performed using main commercial software, including ANSYS package. However, in most cases to solve nonlinear creep equations, one should apply explicit or implicit methods based on the Euler method of approximation of time-derivatives. The Euler method is sufficiently efficient in terms of random access memory in computations, however this method is cumbersome in computation time and does not always provide a required accuracy for creep deformation computations.The paper offers a finite-element algorithm to solve a three-dimensional problem of thermo creep based on the Runge-Kutta finite-difference schemes of different orders with respect to time. It shows a numerical test example to solve the problem on the thermo creep of a beam under tensile loading. The computed results demonstrate that using the Runge-Kutta method with increasing accuracy order allows us to obtain a more accurate solution (with increasing accuracy order by 1 a relative error decreases, approximately, by an order too. The developed algorithm proves to be efficient enough and can be recommended for solving the more complicated problems of thermo creep of structures.

  17. Track parameter propagation through the application of a new adaptive Runge-Kutta-Nystrom method in the ATLAS experiment

    Lund, E; Gavrilenko, I; Strandlie, A


    In this paper we study several fixed step and adaptive Runge-Kutta methods suitable for transporting track parameters through an inhomogeneous magnetic field. Moreover, we present a new adaptive Runge-Kutta-Nystrom method which estimates the local error of the extrapolation without introducing extra stages to the original Runge-Kutta-Nystrom method. Furthermore, these methods are compared for propagation accuracy and computing cost efficiency in the simultaneous track and error propagation (STEP) algorithm of the common ATLAS tracking software. The tests show the new adaptive Runge-Kutta-Nystrom method to be the most computing cost efficient.

  18. On the generation of P-stable exponentially fitted Runge-Kutta-Nystrom methods by exponentially fitted Runge-Kutta methods

    van de Vyver, Hans


    This paper provides an investigation of the stability properties of a family of exponentially fitted Runge-Kutta-Nystrom (EFRKN) methods. P-stability is a very important property usually demanded for the numerical solution of stiff oscillatory second-order initial value problems. P-stable EFRKN methods with arbitrary high order are presented in this work. We have proved our results based on a symmetry argument.

  19. A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge-Kutta method

    Dehghan, Mehdi; Mohammadi, Vahid


    In this research, we investigate the numerical solution of nonlinear Schrödinger equations in two and three dimensions. The numerical meshless method which will be used here is RBF-FD technique. The main advantage of this method is the approximation of the required derivatives based on finite difference technique at each local-support domain as Ωi. At each Ωi, we require to solve a small linear system of algebraic equations with a conditionally positive definite matrix of order 1 (interpolation matrix). This scheme is efficient and its computational cost is same as the moving least squares (MLS) approximation. A challengeable issue is choosing suitable shape parameter for interpolation matrix in this way. In order to overcome this matter, an algorithm which was established by Sarra (2012), will be applied. This algorithm computes the condition number of the local interpolation matrix using the singular value decomposition (SVD) for obtaining the smallest and largest singular values of that matrix. Moreover, an explicit method based on Runge-Kutta formula of fourth-order accuracy will be applied for approximating the time variable. It also decreases the computational costs at each time step since we will not solve a nonlinear system. On the other hand, to compare RBF-FD method with another meshless technique, the moving kriging least squares (MKLS) approximation is considered for the studied model. Our results demonstrate the ability of the present approach for solving the applicable model which is investigated in the current research work.

  20. Monotonicity Conditions for Multirate and Partitioned Explicit Runge-Kutta Schemes

    Hundsdorfer, Willem


    Multirate schemes for conservation laws or convection-dominated problems seem to come in two flavors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods. It will also be seen that the incompatibility of consistency and mass-conservation holds for ‘genuine’ multirate schemes, but not for general partitioned methods.

  1. Numerical Solution of Fuzzy Differential Equations by Runge-Kutta Verner Method

    T. Jayakumar


    Full Text Available In this paper we study the numerical methods for Fuzzy Differential equations by an application of the Runge-Kutta Verner method for fuzzy differential equations. We prove a convergence result and give numerical examples to illustrate the theory.

  2. On spurious steady-state solutions of explicit Runge-Kutta schemes

    Sweby, P. K.; Yee, H. C.; Griffiths, D. F.


    The bifurcation diagram associated with the logistic equation v sup n+1 = av sup n (1-v sup n) is by now well known, as is its equivalence to solving the ordinary differential equation u prime = alpha u (1-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. Runge-Kutta schemes applied to both the equation u prime = alpha u (1-u) and the cubic equation u prime = alpha u (1-u)(b-u) were studied computationally and analytically and their behavior was contrasted with the explicit Euler scheme. Their spurious fixed points and periodic orbits were noted. In particular, it was observed that these may appear below the linearized stability limits of the scheme and, consequently, computation may lead to erroneous results.


    Ai-guo Xiao


    The main purpose of this paper is to present some convergence results for algebraically stable Runge-Kutta methods applied to some classes of one- and two-parameter multiplystiff singular perturbation problems whose stiffness is caused by small parameters and some other factors. A numerical example confirms our results.

  4. Directly Solving Special Second Order Delay Differential Equations Using Runge-Kutta-Nyström Method

    M. Mechee


    Full Text Available Runge-Kutta-Nyström (RKN method is adapted for solving the special second order delay differential equations (DDEs. The stability polynomial is obtained when this method is used for solving linear second order delay differential equation. A standard set of test problems is solved using the method together with a cubic interpolation for evaluating the delay terms. The same set of problems is reduced to a system of first order delay differential equations and then solved using the existing Runge-Kutta (RK method. Numerical results show that the RKN method is more efficient in terms of accuracy and computational time when compared to RK method. The methods are applied to a well-known problem involving delay differential equations, that is, the Mathieu problem. The numerical comparison shows that both methods are in a good agreement.

  5. Discovery and Optimization of Low-Storage Runge-Kutta Methods


    more stage evaluation than RK4, it takes fewer operations to attain the same error level. This savings is one reason why some low storage methods are an...for automatically generating Runge-Kutta trees, order conditions, and truncation error coefficients,” ACM Transac- tions on Mathematical Software...methods. We then focus on optimizing the truncation error coefficients for LSRK to discover new methods. Reusing the tools from the optimization method, we

  6. Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus

    Uğur Kadak


    Full Text Available Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.


    Cheng-jian Zhang


    This paper first presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDEs). Then the numerical analogous results, of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDEs,are given. In particular, it is shown that the (k, l)-algebraic stability of a RK method for ODEs implies the generalized asymptotic stability and the global stability of the induced NRK method.

  8. Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

    Wei-peng HU; Zi-chen DENG; Song-mei HAN; Wei FAN


    Nonlinear wave equations have been extensively investigated in the last several decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation,is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations (PDEs) derived from the Landau-Ginzburg-Higgs equation. The numerical resuits for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.

  9. Three-stage Stiffly Accurate Runge-Kutta Methods for Stiff Stochastic Differential Equations



    In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations (SDEs). Two methods, a three-stage stiffly accurate semi-implicit (SASI3) method and a three-stage stiffly accurate diagonally implicit (SADI3) method, are constructed in this paper. In particular, the truncated random variable is used in the implicit method. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.

  10. a New Methodology for the Construction of Optimized RUNGE-KUTTA-NYSTRÖM Methods

    Papadopoulos, D. F.; Simos, T. E.

    In this paper, a new Runge-Kutta-Nyström method of fourth algebraic order is developed. The new method has zero phase-lag, zero amplification error and zero first integrals of the previous properties. Numerical results indicate that the new method is very efficient for solving numerically the Schrödinger equation. We note that for the first time in the literature we use the requirement of vanishing the first integrals of phase-lag and amplification error in the construction of efficient methods for the numerical solution of the Schrödinger equation.

  11. Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus

    Uğur Kadak; Muharrem Özlük


    Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functio...

  12. Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays

    Chengming HUANG; Stefan VANDEWALLE


    This paper is concerned with the study of the stability of Runge Kutta-Pouzet methods for Volterra integro-differential equations with delays.We are interested in the comparison between the analytical and numerical stability regions.First,we focus on scalar equations with real coefficients.It is proved that all Gauss-Pouzet methods can retain the asymptotic stability of the analytical solution.Then,we consider the multidimensional case.A new stability condition for the stability of the analytical solution is given.Under this condition,the asymptotic stability of Gauss-Pouzet methods is investigated.

  13. Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods

    Puelz, Charles; Canic, Suncica; Rusin, Craig G


    Reduced, or one-dimensional blood flow models take the general form of nonlinear hyperbolic systems, but differ greatly in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we systematically compare several reduced models of blood flow for physiologically relevant vessel parameters, network topology, and boundary data. The models are discretized by a class of Runge-Kutta discontinuous Galerkin methods.

  14. A variable timestep generalized Runge-Kutta method for the numerical integration of the space-time diffusion equations

    Aviles, B.N.; Sutton, T.M.; Kelly, D.J. III.


    A generalized Runge-Kutta method has been employed in the numerical integration of the stiff space-time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic timestep control. The efficiency of the Runge-Kutta method is enhanced by a block-factorization technique that exploits the sparse structure of the matrix system resulting from the space and energy discretized form of the time-dependent neutron diffusion equations. Preliminary numerical evaluation using a one-dimensional finite difference code shows the sparse matrix implementation of the generalized Runge-Kutta method to be highly accurate and efficient when compared to an optimized iterative theta method. 12 refs., 5 figs., 4 tabs.

  15. The numerical solution of differential-algebraic systems by Runge-Kutta methods

    Hairer, Ernst; Lubich, Christian


    The term differential-algebraic equation was coined to comprise differential equations with constraints (differential equations on manifolds) and singular implicit differential equations. Such problems arise in a variety of applications, e.g. constrained mechanical systems, fluid dynamics, chemical reaction kinetics, simulation of electrical networks, and control engineering. From a more theoretical viewpoint, the study of differential-algebraic problems gives insight into the behaviour of numerical methods for stiff ordinary differential equations. These lecture notes provide a self-contained and comprehensive treatment of the numerical solution of differential-algebraic systems using Runge-Kutta methods, and also extrapolation methods. Readers are expected to have a background in the numerical treatment of ordinary differential equations. The subject is treated in its various aspects ranging from the theory through the analysis to implementation and applications.

  16. PIROCK: A swiss-knife partitioned implicit-explicit orthogonal Runge-Kutta Chebyshev integrator for stiff diffusion-advection-reaction problems with or without noise

    Abdulle, Assyr; Vilmart, Gilles


    A partitioned implicit-explicit orthogonal Runge-Kutta method (PIROCK) is proposed for the time integration of diffusion-advection-reaction problems with possibly severely stiff reaction terms and stiff stochastic terms. The diffusion terms are solved by the explicit second order orthogonal Chebyshev method (ROCK2), while the stiff reaction terms (solved implicitly) and the advection and noise terms (solved explicitly) are integrated in the algorithm as finishing procedures. It is shown that the various coupling (between diffusion, reaction, advection and noise) can be stabilized in the PIROCK method. The method, implemented in a single black-box code that is fully adaptive, provides error estimators for the various terms present in the problem, and requires from the user solely the right-hand side of the differential equation. Numerical experiments and comparisons with existing Chebyshev methods, IMEX methods and partitioned methods show the efficiency and flexibility of our new algorithm.

  17. Overcoming Geometry-Induced Stiffness with IMplicit-Explicit (IMEX) Runge-Kutta Algorithms on Unstructured Grids with Applications to CEM, CFD, and CAA

    Kanevsky, Alex


    My goal is to develop and implement efficient, accurate, and robust Implicit-Explicit Runge-Kutta (IMEX RK) methods [9] for overcoming geometry-induced stiffness with applications to computational electromagnetics (CEM), computational fluid dynamics (CFD) and computational aeroacoustics (CAA). IMEX algorithms solve the non-stiff portions of the domain using explicit methods, and isolate and solve the more expensive stiff portions using implicit methods. Current algorithms in CEM can only simulate purely harmonic (up to lOGHz plane wave) EM scattering by fighter aircraft, which are assumed to be pure metallic shells, and cannot handle the inclusion of coatings, penetration into and radiation out of the aircraft. Efficient MEX RK methods could potentially increase current CEM capabilities by 1-2 orders of magnitude, allowing scientists and engineers to attack more challenging and realistic problems.

  18. Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

    Kennedy, Christopher A.; Carpenter, Mark H.


    A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.

  19. The design and applications of Runge-Kutta methods for the simulation of planetary orbits

    Rabbi, S. M. Fajlay

    Since the merger of physics and mathematics at the beginning of 1800s, system of finding solution to n-body problem has been intriguing mathematicians. The resulting differential equations can be solved by a variety of approaches -- for example, the Runge-Kutta Methods (RKn). In this thesis, after a brief historical overview of planetary science, RK3 methods are derived as a three-parameter family of solution methods. A particular instance of this family, FR3, is generated and subsequently tested to show it is indeed a third-order method. The planetary system is modeled as a system of differential of equations using laws of classical mechanics, and the models of planetary motions are generated applying RK4 methods. Kepler's laws of planetary motion are proved empirically using observed data taken from NASA. A new way of expressing Kepler's third law is presented: the orbital velocity of a planet decreases as inverse square root of its orbital radius. Simulation of Sun-Earth-Moon as well as solar system is conducted and compared to that of Dahir's and found is a very similar result. Also, the result of the entire solar system simulation closely matches to that of NASA. Initial position-velocity vectors are generated from NASA-JPL's ephemeris data using post-processing codes obtained from the University of Colorado.

  20. A comparison of Runge-Kutta modifications

    Praagman; N.; Vorst; J.van der; Koster; J.


    Vijf klassen van Runge-Kutta methoden voor het numeriek integreren van beginwaardeproblemen worden vergeleken. Aangetoond wordt dat de beste resultaten, bedoeld is het kleinste aantal rechterlid bewerkingen bij een gegeven tolerantie, verkregen worden met een blok Runge-Kutta methode met een

  1. A combined application of boundary-element and Runge-Kutta methods in three-dimensional elasticity and poroelasticity

    Igumnov Leonid


    Full Text Available The report presents the development of the time-boundary element methodology and a description of the related software based on a stepped method of numerical inversion of the integral Laplace transform in combination with a family of Runge-Kutta methods for analyzing 3-D mixed initial boundary-value problems of the dynamics of inhomogeneous elastic and poro-elastic bodies. The results of the numerical investigation are presented. The investigation methodology is based on direct-approach boundary integral equations of 3-D isotropic linear theories of elasticity and poroelasticity in Laplace transforms. Poroelastic media are described using Biot models with four and five base functions. With the help of the boundary-element method, solutions in time are obtained, using the stepped method of numerically inverting Laplace transform on the nodes of Runge-Kutta methods. The boundary-element method is used in combination with the collocation method, local element-by-element approximation based on the matched interpolation model. The results of analyzing wave problems of the effect of a non-stationary force on elastic and poroelastic finite bodies, a poroelastic half-space (also with a fictitious boundary and a layered half-space weakened by a cavity, and a half-space with a trench are presented. Excitation of a slow wave in a poroelastic medium is studied, using the stepped BEM-scheme on the nodes of Runge-Kutta methods.

  2. Improving the accuracy of simulation of radiation-reaction effects with implicit Runge-Kutta-Nyström methods.

    Elkina, N V; Fedotov, A M; Herzing, C; Ruhl, H


    The Landau-Lifshitz equation provides an efficient way to account for the effects of radiation reaction without acquiring the nonphysical solutions typical for the Lorentz-Abraham-Dirac equation. We solve the Landau-Lifshitz equation in its covariant four-vector form in order to control both the energy and momentum of radiating particles. Our study reveals that implicit time-symmetric collocation methods of the Runge-Kutta-Nyström type are superior in accuracy and better at maintaining the mass-shell condition than their explicit counterparts. We carry out an extensive study of numerical accuracy by comparing the analytical and numerical solutions of the Landau-Lifshitz equation. Finally, we present the results of the simulation of particle scattering by a focused laser pulse. Due to radiation reaction, particles are less capable of penetrating into the focal region compared to the case where radiation reaction is neglected. Our results are important for designing forthcoming experiments with high intensity laser fields.

  3. A New Family of Phase-Fitted and Amplification-Fitted Runge-Kutta Type Methods for Oscillators

    Zhaoxia Chen


    Full Text Available In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta (RK methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase-fitted and amplification-fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.

  4. Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations

    Kasim Hussain


    Full Text Available We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs of the form y(iv=f(x,y denoted as RKFD methods. The first pair, which we will call RKFD5(4, has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5. The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs and solving them.

  5. Numerically optimal Runge-Kutta pairs with interpolants

    Verner, J.


    Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two criteria for selection are proposed with a view to deriving pairs of all orders 6(5) to 9(8) which minimize computation while achieving a user-specified accuracy. Coefficients of improved pairs, their stability regions and coefficients of appended optimal interpolatory Runge-Kutta formulas are provided on the author's website ( This note reports results of tests on these pairs to illustrate their effectiveness in solving nonstiff initial value problems. These pairs and interpolants may be used for implementation, or else to provide comparison targets for other new types of methods such as explicit general linear methods.

  6. Construction and Analysis of Multi-Rate Partitioned Runge-Kutta Methods


    xiii LIST OF ACRONYMS, ABBREVIATIONS, AND TERMS ABP Adams-Bashforth method of order p BDFP Backwards Differentiation Formula of order p...the Adams methods. 1. Adams Methods Within the Adams family of multi-step methods, the two most commonly used are Adams-Bashforth of order p, ( ABP ...look at ABP , as these methods are explicit in time, whereas the AMP are all implicit in time. The general formula for the Adams-Bashforth method

  7. All-stages-implicit and strong-stability-preserving implicit-explicit Runge-Kutta time discretization schemes for hyperbolic systems with stiff relaxation terms

    Duan, Shu-Chao


    We construct eight implicit-explicit (IMEX) Runge-Kutta (RK) schemes up to third order of the type in which all stages are implicit so that they can be used in the zero relaxation limit in a unified and convenient manner. These all-stages-implicit (ASI) schemes attain the strong-stability-preserving (SSP) property in the limiting case, and two are SSP for not only the explicit part but also the implicit part and the entire IMEX scheme. Three schemes can completely recover to the designed accuracy order in two sides of the relaxation parameter for both equilibrium and non-equilibrium initial conditions. Two schemes converge nearly uniformly for equilibrium cases. These ASI schemes can be used for hyperbolic systems with stiff relaxation terms or differential equations with some type constraints.

  8. The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method

    D. F. Papadopoulos


    Full Text Available A new modified Runge-Kutta-Nyström method of fourth algebraic order is developed. The new modified RKN method is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives. Numerical results indicate that the new modified method is much more efficient than other methods derived for solving numerically the Schrödinger equation.

  9. Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

    Yanping Yang


    Full Text Available The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.

  10. Numerical Solution of First-Order Linear Differential Equations in Fuzzy Environment by Runge-Kutta-Fehlberg Method and Its Application

    Sankar Prasad Mondal


    Full Text Available The numerical algorithm for solving “first-order linear differential equation in fuzzy environment” is discussed. A scheme, namely, “Runge-Kutta-Fehlberg method,” is described in detail for solving the said differential equation. The numerical solutions are compared with (i-gH and (ii-gH differential (exact solutions concepts system. The method is also followed by complete error analysis. The method is illustrated by solving an example and an application.

  11. Highly efficient strong stability preserving Runge-Kutta methods with Low-Storage Implementations

    Ketcheson, David I.


    Strong stability-preserving (SSP) Runge–Kutta methods were developed for time integration of semidiscretizations of partial differential equations. SSP methods preserve stability properties satisfied by forward Euler time integration, under a modified time-step restriction. We consider the problem of finding explicit Runge–Kutta methods with optimal SSP time-step restrictions, first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations. By using alternate formulations of the associated optimization problems and introducing a new, more general class of low-storage implementations of Runge–Kutta methods, new optimal low-storage methods and new low-storage implementations of known optimal methods are found. The results include families of low-storage second and third order methods that achieve the maximum theoretically achievable effective SSP coefficient (independent of stage number), as well as low-storage fourth order methods that are more efficient than current full-storage methods. The theoretical properties of these methods are confirmed by numerical experiment.

  12. Adaptive Optimal -Stage Runge-Kutta Methods for Solving Reaction-Diffusion-Chemotaxis Systems

    Jui-Ling Yu


    time step sizes are given explicitly. Yet, theorems about stability and convergence of the algorithm are provided in analyzing robustness and efficiency. Numerical experiment results on a testing problem and a real application problem are shown.

  13. Runge-Kutta-Fehlberg方法在负阻振荡电路求解中的应用%The Application of Runge-Kutta-Fehlberg Method in Solving Negative Resistance Oscillator Circuit

    姚齐国; 李林



  14. Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

    Ademir L. Xavier Jr.


    at once Kirchhoff and filament reference system equations under appropriate initial boundary conditions. To show the application of the method, we display several numerical solutions for filaments including cases showing the effect of gravity.

  15. Symplectic Runge-Kutta Method for Structural Dynamics%结构动力学方程的辛RK方法

    郭静; 邢誉峰


    针对有阻尼和外载荷的线性动力学常微分方程,给出了s级2s阶隐式Gauss-Legendre辛RK(Gauss-Legendre symplectic Runge-Kutta,GLSRK)方法的一种显式高效的执行格式,首次给出了Gauss-Legendre辛RK方法和经典RK方法(classical RK,CRK)的谱半径和单步相位误差的显式表达式,并将两者进行了比较.线性多自由度系统和非线性Rayleigh系统数值算例表明,对结构动力学系统而言,辛RK方法远比经典RK方法优越,在运动学特性和长时间数值模拟方面尤为明显.

  16. On the efficiency of Runge-Kutta-Nystrom methods with interpolants for solving equations of the form Y double prime = F(T,Y,Y prime) over short timespans

    Tsitouras, Ch.; Papageorgiou, G.; Kalvouridis, T.


    Runge-Kutta-Nystrom (RKN) codes for the solution of the initial value problem for the general second-order differential system were developed recently, although the methodology on which they are based was known many years ago. The efficiency of several general Runge-Kutta-Nystrom (GRKN) methods is examined by posing some criteria of cost and accuracy. These methods supplied with the corresponding interpolants are applied to some problems of celestial dynamics. The results obtained show that these codes have good responses in the approximation of the solution of these problems.

  17. Approximating Runge-Kutta matrices by triangular matrices

    W. Hoffmann; J.J.B. de Swart (Jacques)


    textabstractThe implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit




    A nonlinear system with 3 equations and 3 unknowns was got by using symplectic conditions to reduce the system with 8 equations and 4 unknowns, which the coefficients of 4-stage and 4-order diagonally implicit symplectic Runge-Kutta methods must satisfy. An optimal problem was constructed from the nonlinear system. We investigated on the minimum points of the optimal problem and obtained 9 approximate of them. The 9 computational solutions are obtaind respectively,when Broyden-Flecher-Shanno quasi-Newton methods for solve nonlinear equations was used. These solutions can be regarded as the coefficients of fourth-stage and fourth-order diagonally implicit Runge-Kutta methods respectively.

  19. A Runge-Kutta Nystrom algorithm.

    Bettis, D. G.


    A Runge-Kutta algorithm of order five is presented for the solution of the initial value problem where the system of ordinary differential equations is of second order and does not contain the first derivative. The algorithm includes the Fehlberg step control procedure.

  20. Runge Kutta Algorithm applied to a Hydrology Problem

    Narayanan, M.


    In this paper, the author utilizes a fourth order Runge Kutta Algorithm technique to solve a design problem in Hydrology and Fluid Mechanics. Principles of Fuzzy Logic Design methodologies were utilized to analyze the problem and arrive at an appropriate solution. The problem posed was to examine the depletion of water from a reservoir. A suitable model was to be created to represent different parameters that contributed to the depletion, such as evaporation, drainage and seepage, irrigation channels, city water supply pipes, etc. The reservoir was being fed via natural resources such as rain, streams, rivers, etc. A model of a catchment area and a reservoir lake is simulated as a tank and exit discharge is represented as fluid output via a long pipe. The Input to the reservoir is assumed to be continuous-time and time varying. In other words, the flow rate of fluid input is presumed to change with time. The required objective is to maintain a predetermined level of water in the reservoir, regardless of input conditions. This is accomplished by adjusting the depletion rate. This means that some of the Irrigation channels may have to be closed or some of the city water supply lines need to be shut off. The differential equation governing the system can be easily derived using Bernoulli's' equation. If hd is the desired height of water in the reservoir and h(t) represents the height of water in the reservoir at any given time, K represents a positive constant. (dh/dt) + K [ h(t) - hd ] = 0 The closed loop system is simulated by using fourth-order Runge-Kutta algorithm. The controller output u(t) can be calculated using the above equation. The Runge-Kutta algorithm is a very popular method, which is widely used for obtaining a numerical solution to a given differential equation. The Runge-Kutta algorithm is considered to be quite accurate for a broad range of scientific and engineering applications, and as such, the method is heavily used by many scholars and

  1. Application of Runge Kutta time marching scheme for the computation of transonic flows in turbomachines

    Subramanian, S. V.; Bozzola, R.


    Numerical solutions of the unsteady Euler equations are obtained using the classical fourth order Runge Kutta time marching scheme. This method is fully explicit and is applied to the governing equations in the finite volume, conservation law form. In order to determine the efficiency of this scheme for solving turbomachinery flows, steady blade-to-blade solutions are obtained for compressor and turbine cascades under subsonic and transonic flow conditions. Computed results are compared with other numerical methods and wind tunnel measurements. The study also focuses on other important numerical aspects influencing the performance of the algorithm and the solution accuracy such as grid types, boundary conditions and artificial viscosity. For this purpose, H, O, and C type computational grids as well as characteristic and extrapolation type boundary conditions are included in solution procedures.

  2. A class of high-order Runge-Kutta-Chebyshev stability polynomials

    O'Sullivan, Stephen


    The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L = MN$ are used to construct explicit schemes comprising $L$ forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to $\\sim L^2$. The associated stability domain scales as $M^2$ along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure. By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher group composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKC schemes are efficient ...

  3. Application of Runge Kutta time marching scheme for the computation of transonic flows in turbomachines

    Subramanian, S. V.; Bozzola, R.


    Numerical solutions of the unsteady Euler equations are obtained using the classical fourth order Runge Kutta time marching scheme. This method is fully explicit and is applied to the governing equations in the finite volume, conservation law form. In order to determine the efficiency of this scheme for solving turbomachinery flows, steady blade-to-blade solutions are obtained for compressor and turbine cascades under subsonic and transonic flow conditions. Computed results are compared with other numerical methods and wind tunnel measurements. The present study also focuses on other important numerical aspects influencing the performance of the algorithm and the solution accuracy such as grid types, boundary conditions, and artificial viscosity. For this purpose, H, O, and C type computational grids as well as characteristic and extrapolation type boundary conditions are included in the solution procedure.

  4. Explicit strong stability preserving multistep Runge–Kutta methods

    Bresten, Christopher


    High-order spatial discretizations of hyperbolic PDEs are often designed to have strong stability properties, such as monotonicity. We study explicit multistep Runge-Kutta strong stability preserving (SSP) time integration methods for use with such discretizations. We prove an upper bound on the SSP coefficient of explicit multistep Runge-Kutta methods of order two and above. Numerical optimization is used to find optimized explicit methods of up to five steps, eight stages, and tenth order. These methods are tested on the linear advection and nonlinear Buckley-Leverett equations, and the results for the observed total variation diminishing and/or positivity preserving time-step are presented.

  5. Stochastic Runge-Kutta Software Package for Stochastic Differential Equations

    Gevorkyan, M N; Korolkova, A V; Kulyabov, D S; Sevastyanov, L A


    As a result of the application of a technique of multistep processes stochastic models construction the range of models, implemented as a self-consistent differential equations, was obtained. These are partial differential equations (master equation, the Fokker--Planck equation) and stochastic differential equations (Langevin equation). However, analytical methods do not always allow to research these equations adequately. It is proposed to use the combined analytical and numerical approach studying these equations. For this purpose the numerical part is realized within the framework of symbolic computation. It is recommended to apply stochastic Runge--Kutta methods for numerical study of stochastic differential equations in the form of the Langevin. Under this approach, a program complex on the basis of analytical calculations metasystem Sage is developed. For model verification logarithmic walks and Black--Scholes two-dimensional model are used. To illustrate the stochastic "predator--prey" type model is us...

  6. Implicit - symplectic partitioned (IMSP) Runge-Kutta schemes for predator-prey dynamics

    Diele, F.; Marangi, C.; Ragni, S.


    In the study of the effects of habitat fragmentation on biodiversity the role of spatial processes reveals of great interest since both the variation of size of the domains as well as their heterogeneity largely affects the dynamics of species. In order to begin a preliminary study about the effects of habitat fragmentation on wolf - wild boar pair populating the Italian "Alta Murgia" Natura 2000 site, object of interest for FP7 project BIOSOS, (BIOdiversity multi-SOurce Monitoring System: from Space TO Species), spatially explicit models described by reaction-diffusion partial differential equations are considered. Numerical methods based on partitioned Runge-Kutta schemes which use an implicit scheme for the stiff diffusive term and a partitioned symplectic scheme for the reaction function are here proposed. We are motivated by the classical results about Lotka-Volterra model described by ordinary differential equations to which the spatially explicit model reduces for diffusion coefficients tending to zero: for their accurate solution symplectic schemes have to be used for an optimal long run preservation of the dynamics invariant. Moreover, for models based on logistic growth and Holling type II functional predator response we verify the better performance of our schemes when compared with classical implicit-explicit (IMEX) schemes on chaotic dynamics given in literature.

  7. Analytical Benchmarking, Precision Particle Tracking, Electric and Magnetic Storage Rings, Runge-Kutta, Predictor-Corrector

    Metodiev, E M; Fandaros, M; Haciomeroglu, S; Huang, D; Huang, K L; Patil, A; Prodromou, R; Semertzidis, O A; Sharma, D; Stamatakis, A N; Orlov, Y F; Semertzidis, Y K


    A set of analytical benchmarks for tracking programs are required for precision storage ring experiments. To determine the accuracy of precision tracking programs in electric and magnetic rings, a variety of analytical estimates of particle and spin dynamics in the rings are developed and compared to the numerical results of tracking simulations. Initial discrepancies in the comparisons indicated the need for improvement of several of the analytical estimates. As an example, we find that the fourth order Runge-Kutta/Predictor-Corrector method was accurate but slow, and that it passed all the benchmarks it was tested against, often to the sub-part per billion level. Thus high precision analytical estimates and tracking programs based on fourth order Runge-Kutta/Predictor-Corrector integration can be used to benchmark faster tracking programs for accuracy.

  8. Simulation of manufacturing sequences using higher order accurate Runge-Kutta time integration schemes

    Gleim, Tobias; Schröder, Bettina; Kuhl, Detlef


    This paper deals with the numerical simulation of multi-field problems in the context of functionally graded materials. The corresponding manufacturing sequences are mostly characterized by strong interacting fields with different physical behaviors, which additionally have high dynamic responses. In order to solve these distinct processes with a high accuracy in the time, various RUNGE-KUTTA methods are investigated. Furthermore, a h-error estimator and an embedded error estimator are considered for a qualitative evaluation of the results.

  9. Another approach to Runge-Kutta methods

    Traas, C.R.


    The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct optimization, avoiding the necessity to introduce simplifying

  10. A generalization of the Runge-Kutta iteration

    Haelterman, R.; Vierendeels, J.; van Heule, D.


    Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge-Kutta iteration. We show that a generally used formulation [A. Jameson, Numerical solution of the Euler equations for compressible inviscid fluids, in: F. Angrand, A. Dervieux, J.A. Désidéri, R. Glowinski (Eds.), Numerical Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, pp. 199-245] does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this, without using an excessive number of coefficients. After having converted the optimal parameters found in previous studies (e.g. [B. Van Leer, C.H. Tai, K.G. Powell, Design of optimally smoothing multi-stage schemes for the Euler equations, AIAA Paper 89-1923, 1989]) we compare them with those that we obtain when we optimize for an integrated 2-grid V-cycle and show that this results in superior performance using a low number of stages. We also propose a variant of our new formulation that roughly follows the idea of the Martinelli-Jameson scheme [A. Jameson, Analysis and design of numerical schemes for gas dynamics 1, artificial diffusion, upwind biasing, limiter and their effect on multigrid convergence, Int. J. Comput. Fluid Dyn. 4 (1995) 171-218; J.V. Lassaline, Optimal multistage relaxation coefficients for multigrid flow solvers.] used on the advection-diffusion equation which that can be extended to other types. Gains in the order of 30%-50% have been shown with respect to classical iterative schemes on the advection equation. Better results were also obtained on the advection-diffusion equation than with the Martinelli-Jameson coefficients, but with less than half the number of matrix-vector multiplications.

  11. A note on semi-discrete conservation laws and conservation of wave action by multisymplectic Runge-Kutta box schemes

    Frank, J.E.


    In this note we show that multisymplectic Runge-Kutta box schemes, of which the Gauss-Legendre methods are the most important, preserve a discrete conservation law of wave action. The result follows by loop integration over an ensemble of flow realizations, and the local energy-momentum conservation

  12. A Study on Third Order Runge-Kutta Techniques for Solving Practical Problems



    Full Text Available In this paper, an analysis has been carried out to examine Nystrom third order, Heun third order and Classical Runge-Kutta third order methods to solve image processing and numerical problems which are demonstrated in brief. The methods adapted are fully capable to cope with the linearity and nonlinearity of the physical problems with versatile physical nature. Example problems and its corresponding results are exhibited which reveal the efficiency and reliability of the employed techniques. Furthermore, validity of an obtained solution is verified in comparison with the simulation output for an image processing problem and numerically computed results for an engineering problem and initial value problems.

  13. 两步龙格库塔方法对多延迟量微分代数方程的渐近稳定性%Asymptotic stability of two-step Runge-Kutta methods for differential-algebraic equations with several delays

    李晓燕; 孙乐平; 毛宏坤


    The two-step Runge-Kutta methods for the differential-algebraic equations with several delays are developed and it is proved that the methods are asymptotically stable under the assumption that the coefficient matrices are all upper triangular. This assumption is regarded as true for DDAEs which have a wide range of applications.%研究了用两步龙格库塔方法求解多延迟微分代数方程的渐进稳定性,并且证明了在微分代数方程矩阵都是上三角矩阵的假设下,两步龙格库塔法求解此类方程是渐进稳定的.这种假设对于有广泛应用的海参伯格微分代数方程是正确的.

  14. A four-stage fourth order Runge-Kutta time high-order pseudospectral method for acoustic equation simulation%一种四级四阶龙格-库塔时间高阶伪谱法声波方程模拟

    杨怀英; 唐小平; 刘宽厚


    The Runge-Kutta method is a common technology for solving the Ordinary Differential Equations ( ODE) and is characterized by high precision, strong stability and some other advantages. In this paper, based on a new four-stage fourth order Runge-Kutta meth-od, the authors first combined the four-stage calculation formula with a new two-stage iteration formula, thus achieving the purpose of saving computational memory. And then, the time high-order discrete form of the acoustic wave equation was derived and, in combina-tion with the pseudospectral method, some researches on the high-accuracy and high-definition acoustic wave field simulation technology of the four-stage fourth order Runge-Kutta time high-order pseudospectral method were carried out, with an investigation of the stability and dispersion of the method. Finally, homogeneous media, layered media and lens model were selected for wave field simulation test. The simulation results show that the four-stage fourth order Runge-Kutta time high-order pseudospectral method has strong stability and high wave field definition and can effectively remove the dispersion and adapt itself to large simulation parameter range, thus being a high efficient wave field simulation method with great application potential.%龙格-库塔法是常用于求解常微分方程( ODE)的一项技术,该技术具有精度高、稳定性强等特点。笔者以一种新的四级四阶龙格-库塔法为基础,先将其四级计算公式合并为新的两级计算迭代公式,从而达到节约计算内存的目的;再以此为基础推导出声波方程的时间高阶离散形式,并与伪谱法技术相结合,研究四级四阶龙格-库塔时间高阶伪谱法声波高精度、高清晰度的波场模拟技术,进而研究该方法的稳定性与频散特性;最后,分别选取均匀介质、层状介质和透镜体模型进行波场模拟试验。模拟结果表明,该方法具有稳定性强、能有效去除

  15. Numerical Stability of Runge-Kutta Methods of the Functional Multi-delay Differential Equations with Piecewise Continuous Arguments%分段连续型混合泛函多延迟微分方程Runge-Kutta方法的数值稳定性



    By application of Runge-Kutta methods to solving the functional multi-delay differential equations with piecewise continuous arguments,the conditions under which the numerical solution is asymptotically stable are obtained.By means of the theory of Padé approximation,the necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are obtained and some numerical experiments are given.%将Runge-Kutta方法用于求解自变量分段连续型混合泛函多延迟微分方程,得到了数值解渐近稳定的条件.利用Padé逼近理论得到了数值解的渐近稳定区域包含解析解的渐近稳定区域的充分必要条件,并给出了几个数值算例.

  16. Exponential Runge-Kutta schemes for inhomogeneous Boltzmann equations with high order of accuracy

    Li, Qin


    We consider the development of exponential methods for the robust time discretization of space inhomogeneous Boltzmann equations in stiff regimes. Compared to the space homogeneous case, or more in general to the case of splitting based methods, studied in Dimarco Pareschi (SIAM J. Num. Anal. 2011) a major difficulty is that the local Maxwellian equilibrium state is not constant in a time step and thus needs a proper numerical treatment. We show how to derive asymptotic preserving (AP) schemes of arbitrary order and in particular using the Shu-Osher representation of Runge-Kutta methods we explore the monotonicity properties of such schemes, like strong stability preserving (SSP) and positivity preserving. Several numerical results confirm our analysis.

  17. Runge-Kutta model-based nonlinear observer for synchronization and control of chaotic systems.

    Beyhan, Selami


    This paper proposes a novel nonlinear gradient-based observer for synchronization and observer-based control of chaotic systems. The model is based on a Runge-Kutta model of the chaotic system where the evolution of the states or parameters is derived based on the error-square minimization. The stability and convergence conditions of observer and control methods are analyzed using a Lyapunov stability approach. In numerical simulations, the proposed observer and well-known sliding-mode observer are compared for the synchronization of a Lü chaotic system and observer-based stabilization of a Chen chaotic system. The noisy case for synchronization and parameter uncertainty case for stabilization are also considered for both observer-based methods. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.

  18. High Order Adjoint Derivatives using ESDIRK Methods for Oil Reservoir Production Optimization

    Capolei, Andrea; Stenby, Erling Halfdan; Jørgensen, John Bagterp


    In production optimization, computation of the gradients is the computationally expensive step. We improve the computational efficiency of such algorithms by improving the gradient computation using high-order ESDIRK (Explicit Singly Diagonally Implicit Runge-Kutta) temporal integration methods...

  19. Algebraic dynamics algorithm: Numerical comparison with Runge-Kutta algorithm and symplectic geometric algorithm

    WANG ShunJin; ZHANG Hua


    Based on the exact analytical solution of ordinary differential equations,a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm.A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models.The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision,and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.

  20. Algebraic dynamics algorithm:Numerical comparison with Runge-Kutta algorithm and symplectic geometric algorithm


    Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.

  1. On Error Estimation in Runge-Kutta Methods

    Gbolahan BOLARIN


    Full Text Available The increase in PCs' capabilities and communication bandwidth over the last decade has made distributed computing a more practical idea for solving computational problems. We have developed a decentralized P2P system called ParCop (Parallel Cooperation. ParCop enables each peer in a P2P network to view the rest of the network as a supercomputer, by running ParCop system software on the machine as a daemon service. ParCop allows participants to execute different applications on shared resources owned by other participants. In this paper, we present the new capabilities of ParCop system: efficient resource discovery by using the Blackboard Resource Discovery Mechanism (BRDM, adaptation in dynamic networks, effective data caching, efficient scaling and the provision of a secure environment. We also present three scheduling policies that allow peers in ParCop environment to take scheduling decisions based on the information coming from the peers in the network. The use of these scheduling policies minimizes the processing time of applications in ParCop, improves the ability of dealing with peers which have different capabilities and requirements, and achieves efficient load balancing.

  2. Optimal explicit strong stability preserving Runge–Kutta methods with high linear order and optimal nonlinear order

    Gottlieb, Sigal


    High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge-Kutta methods exist only up to fourth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: explicit SSP Runge-Kutta methods of any linear order exist. These methods reduce to second order when applied to nonlinear problems. In the current work we aim to find explicit SSP Runge-Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. These methods have strong stability coefficients that approach those of the linear methods as the number of stages and the linear order is increased. This work shows that when a high linear order method is desired, it may still be worthwhile to use methods with higher nonlinear order.

  3. Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

    Abdulle, Assyr


    We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the step size reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta-Chebyshev (ROCK2) methods for deterministic problems. The convergence, meansquare, and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results. © 2013 Society for Industrial and Applied Mathematics.

  4. Equations of condition for high order Runge-Kutta-Nystrom formulae

    Bettis, D. G.


    Derivation of the equations of condition of order eight for a general system of second-order differential equations approximated by the basic Runge-Kutta-Nystrom algorithm. For this general case, the number of equations of condition is considerably larger than for the special case where the first derivative is not present. Specifically, it is shown that, for orders two through eight, the number of equations for each order is 1, 1, 1, 2, 3, 5, and 9 for the special case and is 1, 1, 2, 5, 13, 34, and 95 for the general case.

  5. A new explicit method for the numerical solution of parabolic differential equations

    Satofuka, N.


    A new method is derived for solving parabolic partial differential equations arising in transient heat conduction or in boundary-layer flows. The method is based on a combination of the modified differential quadrature (MDQ) method with the rational Runge-Kutta time-integration scheme. It is fully explicit, requires no matrix inversion, and is stable for any time-step for the heat equations. Burgers equation and the one- and two-dimensional heat equations are solved to demonstrate the accuracy and efficiency of the proposed algorithm. The present method is found to be very accurate and efficient when results are compared with analytic solutions.

  6. Gauss collocation methods for efficient structure preserving integration of post-Newtonian equations of motion

    Seyrich, Jonathan


    In this work, we present the hitherto most efficient and accurate method for the numerical integration of post-Newtonian equations of motion. We first transform the Poisson system as given by the post-Newtonian approximation to canonically symplectic form. Then we apply Gauss Runge-Kutta schemes to numerically integrate the resulting equations. This yields a convenient method for the structure preserving long-time integration of post-Newtonian equations of motion. In extensive numerical experiments, this approach turns out to be faster and more accurate i) than previously proposed structure preserving splitting schemes and ii) than standard explicit Runge-Kutta methods.

  7. Single Shooting and ESDIRK Methods for adjoint-based optimization of an oil reservoir


    are large-scale problems and require specialized numerical algorithms. In this paper, we combine a single shooting optimization algorithm based on sequential quadratic programming (SQP) with explicit singly diagonally implicit Runge-Kutta (ESDIRK) integration methods and a continuous adjoint method...

  8. Variational formulation of the method of lines and its application to the wave propagation problems

    Shatalov, M


    Full Text Available = = to the system of ordinary differential equations (3.5) the initial problem will be formulated and could be solved by one of the available numerical methods (Runge-Kutta, Adams, etc.). Explicit form of the system (3.5) is as follows: ( ) ( ) ( ) ( ) 2 2...

  9. Solution of Constrained Optimal Control Problems Using Multiple Shooting and ESDIRK Methods

    Capolei, Andrea; Jørgensen, John Bagterp


    In this paper, we describe a novel numerical algorithm for solution of constrained optimal control problems of the Bolza type for stiff and/or unstable systems. The numerical algorithm combines explicit singly diagonally implicit Runge-Kutta (ESDIRK) integration methods with a multiple shooting...

  10. A stabilized explicit Lagrange multiplier based domain decomposition method for parabolic problems

    Zheng, Zheming; Simeon, Bernd; Petzold, Linda


    A fully explicit, stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations (PDEs) is presented. Writing the semi-discretized equations as a differential-algebraic equation (DAE) system where the interface continuity constraints between subdomains are enforced by Lagrange multipliers, the method uses the Runge-Kutta-Chebyshev projection scheme to integrate the DAE explicitly and to enforce the constraints by a projection. With mass lumping techniques and node-to-node matching grids, the method is fully explicit without solving any linear system. A stability analysis is presented to show the extended stability property of the method. The method is straightforward to implement and to parallelize. Numerical results demonstrate that it has excellent performance.

  11. 12 Steps Runge-kutta 2 Orders Algorithm for Satellite Orbit Integration%12阶 Runge-kutta 2次算法的卫星轨道积分研究

    李得海; 袁运斌; 欧吉坤; 闫伟


    研究了12阶Runge-kutta 2次算法由加速度直接积分位置得到卫星轨道,并将其应用于人造卫星轨道积分.实验结果表明,与传统单步法、同阶多步法相比,12阶Runge-kutta 2次算法在积分精度和稳定性方面具有明显的优势,但相同步长下较其他方法计算耗时多,运算复杂.综合考虑,可以利用其积分误差随步长增加而维持稳定的特点,通过适当增加步长降低计算耗时,满足高轨卫星轨道预报与精密定轨的应用需求.

  12. Classical seventh-, sixth-, and fifth-order Runge-Kutta-Nystrom formulas with stepsize control for general second-order differential equations

    Fehlberg, E.


    Runge-Kutta-Nystrom formulas of the seventh, sixth, and fifth order were derived for the general second order (vector) differential equation written as the second derivative of x = f(t, x, the first derivative of x). The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the local truncation error in x, and they require no more evaluations per step than the earlier Runge-Kutta formulas for the first derivative of x = f(t, x). The developed formulas are expected to be time saving in comparison to the Runge-Kutta formulas for first-order differential equations, since it is not necessary to convert the second-order differential equations into twice as many first-order differential equations. The examples shown saved from 25 percent to 60 percent more computer time than the earlier formulas for first-order differential equations, and are comparable in accuracy.

  13. A simplified Nyström-tree theory for extended Runge-Kutta-Nyström integrators solving multi-frequency oscillatory systems

    Yang, Hongli; Zeng, Xianyang; Wu, Xinyuan; Ru, Zhengliang


    In the study of extended Runge-Kutta-Nyström (abbr. ERKN) methods for the integration of multi-frequency oscillatory systems, a quite complicated set of algebraic conditions arises which must be satisfied for a method to achieve some specified order. A theory of tri-colored tree was proposed by Yang et al. (2009), for achieving the order conditions of ERKN methods which are designed specially for multi-frequency and multidimensional perturbed oscillators. The tri-colored tree theory for the order conditions in that paper is useful, but not completely satisfactory due to the existence of redundant trees. In this paper, a simplified tri-colored theory and the order conditions for ERKN integrators are developed by constructing a set of simplified special extended Nyström trees (abbr. SSENT) and defining some real-valued mappings on it. In order to simplify the tri-colored tree theory, two special mappings, the extended elementary differential and the sign mapping for a tree are investigated in detail. This leads to a novel Nyström-tree theory for the order conditions for ERKN methods without any redundant trees, which simplifies the tri-colored theory.

  14. ARK methods up to order five

    Butcher, J. C.


    Almost Runge-Kutta methods (or "ARK methods") have many of the advantages of Runge-Kutta methods but, for many problems, are capable of greater accuracy. In this paper a complete classification of fourth order ARK methods with 4 stages is presented. The paper also analyzes fifth order methods with 5 or with 6 stages. Some limited numerical experiments show that the new methods are capable of excellent performance, comparable to that of known highly efficient Runge-Kutta methods.

  15. ARK methods: some recent developments

    Moir, Nicolette


    Almost Runge-Kutta methods are a sub-class of the family of methods known as general linear methods, used for solving ordinary differential equations. They combine many of the favourable properties of traditional Runge-Kutta methods with some additional advantages. We will introduce these methods, concentrating on methods of order four, and present some recent results.

  16. Poincaré Map Based on Splitting Methods

    GANG Tie-Qiang; CHEN Li-Jie; MEI Feng-Xiang


    Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact flow. However, for the explicit Runge-Kutta methods, there is an error term of order p + 1 for the Jacobian matrix determinants. Then, the volume evolution law of a given region in phase space is discussed for different algorithms. It is proved that splitting methods can exactly preserve the sum of Lyapunov exponents invariable. Finally, a Poincaré map and its energy distribution of the Duffing equation are computed by using the second-order splitting method and the Heun method (a second-order Runge-Kutta method). Computation illustrates that the results by splitting methods can properly represent systems' chaotic phenomena.


    孙建强; 马中骐; 秦孟兆


    A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the RungeKutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.

  18. High-order implicit time-marching methods for unsteady fluid flow simulation

    Boom, Pieter David

    Unsteady computational fluid dynamics (CFD) is increasingly becoming a critical tool in the development of emerging technologies and modern aircraft. In spite of rapid mathematical and technological advancement, these simulations remain computationally intensive and time consuming. More efficient temporal integration will promote a wider use of unsteady analysis and extend its range of applicability. This thesis presents an investigation of efficient high-order implicit time-marching methods for application in unsteady compressible CFD. A generalisation of time-marching methods based on summation-by-parts (SBP) operators is described which reduces the number of stages required to obtain a prescribed order of accuracy, thus improving their efficiency. The classical accuracy and stability theory is formally extended for these generalised SBP (GSBP) methods, including superconvergence and nonlinear stability. Dual-consistent SBP and GSBP time-marching methods are shown to form a subclass of implicit Runge-Kutta methods, which enables extensions of nonlinear accuracy and stability results. A novel family of fully-implicit GSBP Runge-Kutta schemes based on Gauss quadrature are derived which are both algebraically stable and L-stable with order 2s - 1, where s is the number of stages. In addition, a numerical tool is developed for the construction and optimisation of general linear time-marching methods. The tool is applied to the development of several low-stage-order L-stable diagonally-implicit methods, including a diagonally-implicit GSBP Runge-Kutta scheme. The most notable and efficient method developed is a six-stage fifth-order L-stable stiffly-accurate explicit-first-stage singly-diagonally-implicit Runge-Kutta (ESDIRK5) method with stage order two. The theoretical results developed in this thesis are supported by numerical simulations, and the predicted relative efficiency of the schemes is realised.

  19. Explicit high-order symplectic integrators for charged particles in general electromagnetic fields

    Tao, Molei


    This article considers non-relativistic charged particle dynamics in both static and non-static electromagnetic fields, which are governed by nonseparable, possibly time-dependent Hamiltonians. For the first time, explicit symplectic integrators of arbitrary high-orders are constructed for accurate and efficient simulations of such mechanical systems. Performances superior to the standard non-symplectic method of Runge-Kutta are demonstrated on two examples: the first is on the confined motion of a particle in a static toroidal magnetic field used in tokamak; the second is on how time-periodic perturbations to a magnetic field inject energy into a particle via parametric resonance at a specific frequency.

  20. 点隐式龙格-库塔方法的应用研究%Some Investigations on Applications of Point-implicit Runge-Kurtta Method

    李典; 杨永


    为了提高求解器的效率,在显式龙格-库塔时间推进的欧拉方程求解器之上,发展了点隐式龙格-库塔时间推进格式.给出了其推导过程和非结构网格下中心格式和迎风格式(Roe和Van Leer格式)预处理算子的构造方法.NACA0012翼型和RAE2822翼型的跨音速无粘流动模拟表明:与显式龙格-库塔方法相比,方法能提高求解效率且内存需求相当,具有一定的工程应用价值.%In order to improve the efficiency of flow solver ,a point- implicit Runge - Kutta method is raised on the basis of the Euler solver with an explicit Runge- Kutta method time marching.The derivation of preconditioner is presented.Its constructions,according to the central and upwind ( Roe- antl Van Leerscheme) spatial discretization,based on unstructured grid are given,respectively.The simulations of the transonic inviscid flow around NACA0012 and RAE2822 show that comparing with the explicit RungeKutta method,the efficientcy is improved by the presented algorithm with a moderate memory increment.

  1. Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods

    DeBonis, James R.


    A computational fluid dynamics code that solves the compressible Navier-Stokes equations was applied to the Taylor-Green vortex problem to examine the code s ability to accurately simulate the vortex decay and subsequent turbulence. The code, WRLES (Wave Resolving Large-Eddy Simulation), uses explicit central-differencing to compute the spatial derivatives and explicit Low Dispersion Runge-Kutta methods for the temporal discretization. The flow was first studied and characterized using Bogey & Bailley s 13-point dispersion relation preserving (DRP) scheme. The kinetic energy dissipation rate, computed both directly and from the enstrophy field, vorticity contours, and the energy spectra are examined. Results are in excellent agreement with a reference solution obtained using a spectral method and provide insight into computations of turbulent flows. In addition the following studies were performed: a comparison of 4th-, 8th-, 12th- and DRP spatial differencing schemes, the effect of the solution filtering on the results, the effect of large-eddy simulation sub-grid scale models, and the effect of high-order discretization of the viscous terms.

  2. Fermentation Process Modeling with Levenberg-Marquardt Algorithm and Runge-Kutta Method on Ethanol Production by Saccharomyces cerevisiae

    Dengfeng Liu


    Full Text Available The core of the Chinese rice wine making is a typical simultaneous saccharification and fermentation (SSF process. In order to control and optimize the SSF process of Chinese rice wine brewing, it is necessary to construct kinetic model and study the influence of temperature on the Chinese rice wine brewing process. An unstructured kinetic model containing 12 kinetics parameters was developed and used to describe the changing of kinetic parameters in Chinese rice wine fermentation at 22, 26, and 30°C. The effects of substrate and product inhibitions were included in the model, and four variable, including biomass, ethanol, sugar and substrate were considered. The R-square values for the model are all above 0.95 revealing that the model prediction values could match experimental data very well. Our model conceivably contributes significantly to the improvement of the industrial process for the production of Chinese rice wine.

  3. An embedded pair of method of orders 6(4) with 6 stages for special systems of ordinary differential equations

    Olemskoy, I. V.; Eremin, A. S.


    We construct here an embedded Dormand-Prince pair of explicit methods of orders 6 and 4 for systems of ordinary differential equations with special structure, namely with two parts, in which the right-hand sides are dependent only on the unknown functions from the other group. The number of stages is six, which is fewer than for general explicit Runge-Kutta methods. The comparison to Dormand-Prince method of the same computation cost is made showing the higher accuracy of the suggested method.


    Xiao-qiu Song


    In this paper, the theory of parallel multi-stage & multi-step method is dis cussed, which is a form of combining Runge-Kutta method with linear multi-step method that can be used for parallel computation.

  5. Optimal explicit strong-stability-preserving general linear methods : complete results.

    Constantinescu, E. M.; Sandu, A.; Mathematics and Computer Science; Virginia Polytechnic Inst. and State Univ.


    This paper constructs strong-stability-preserving general linear time-stepping methods that are well suited for hyperbolic PDEs discretized by the method of lines. These methods generalize both Runge-Kutta (RK) and linear multistep schemes. They have high stage orders and hence are less susceptible than RK methods to order reduction from source terms or nonhomogeneous boundary conditions. A global optimization strategy is used to find the most efficient schemes that have low storage requirements. Numerical results illustrate the theoretical findings.

  6. A Survey of Symplectic and Collocation Integration Methods for Orbit Propagation

    Jones, Brandon A.; Anderson, Rodney L.


    Demands on numerical integration algorithms for astrodynamics applications continue to increase. Common methods, like explicit Runge-Kutta, meet the orbit propagation needs of most scenarios, but more specialized scenarios require new techniques to meet both computational efficiency and accuracy needs. This paper provides an extensive survey on the application of symplectic and collocation methods to astrodynamics. Both of these methods benefit from relatively recent theoretical developments, which improve their applicability to artificial satellite orbit propagation. This paper also details their implementation, with several tests demonstrating their advantages and disadvantages.

  7. The Harmonious Dissipative Operators and the Completely Square Conservative Difference Scheme in an Explicit Way

    王斌; 季仲贞


    In this paper, a new definition on harmonious dissipative operators is given and some important properties of theirs are shown. Especially, the relationship between a harmonious dissipative operator and the completely square conservative difference scheme in an explicit way is revealed. Kinds of 2-order, 3-order and 4-order harmonious dissipative operators are constructed by using the traditional Runge-Kutta method and a species of general m-order harmonious dissipative operators is established in the linear case. In addition, an efficiency parameter to appraise the time benefits of a harmonious dissipative operator is defined in this paper. It is testified in numerical tests that the harmonious dissipative operators are indeed able to improve the time-efficiency and computational effect of the completely square conservative difference scheme in an explicit way.

  8. A regularized model for impact in explicit dynamics applied to the split Hopkinson pressure bar

    Otto, Peter; De Lorenzis, Laura; Unger, Jörg F.


    In the numerical simulation of impact phenomena, artificial oscillations can occur due to an instantaneous change of velocity in the contact area. In this paper, a nonlinear penalty regularization is used to avoid these oscillations. A particular focus is the investigation of higher order methods in space and time to increase the computational efficiency. The spatial discretization is realized by higher order spectral element methods that are characterized by a diagonal mass matrix. The time integration scheme is based on half-explicit Runge-Kutta scheme of fourth order. For the conditionally stable scheme, the critical time step is influenced by the penalty regularization. A framework is presented to adjust the penalty stiffness and the time step for a specific mesh to avoid oscillations. The methods presented in this paper are applied to 1D-simulations of a split Hopkinson pressure bar, which is commonly used for the investigation of materials under dynamic loading.

  9. Ground p enetrating radar numerical simulation with interp olating wavelet scales metho d and research on fourth-order Runge-Kutta auxiliary differential equation p erfectly matched layer%插值小波尺度法探地雷达数值模拟及四阶Runge Kutta辅助微分方程吸收边界条件∗

    冯德山; 杨道学; 王珣


    应用迭代插值方法构造了插值小波尺度函数,并将该尺度函数的导数用于离散Maxwell方程组的空间微分,使用四阶Runge Kutta(four order Runge Kutta, RK4)算法计算时间导数,导出了插值小波尺度法的探地雷达(ground penetrating radar, GPR)正演公式,与常规的基于中心差分的时域有限差分算法(finite difference time domain, FDTD)相比,插值小波尺度算法提高了GPR波动方程的空间与时间离散精度。首先,采用具有解析解的层状模型,分别将FDTD算法及插值小波尺度法应用于层状模型正演,单道雷达数据与解析解拟合表明:相同的网格剖分方式,插值小波尺度法比FDTD具有更高的精度。然后,将辅助微分方程完全匹配层(auxiliary differential equation perfecting matched layer, ADE-PML)边界条件应用到插值小波尺度法GPR正演中,在均匀介质模型中对比了FDTD-CPML(坐标伸缩完全匹配层), FDTD-RK4ADE-PML、插值小波尺度RK4ADE-PML的反射误差,结果表明:插值小波尺度RK4ADE-PML吸收效果优于另外两种条件下的吸收边界。最后,应用加载UPML(各向异性完全匹配层)的FDTD和RK4ADE-PML的插值小波尺度法开展了二维GPR模型的正演,展示了RK4ADE-PML对倏逝波的良好吸收效果。%Ground penetrating radar (GPR) forward is one of the geophysical research directions. Through the forward of geological model, the database of radar model can be enriched and the characteristics of typical geological radar echo images can be understood, which in turn can guide the data interpretation of GPR measured profile, thereby improving the GPR data interpretation level. In this article, the interpolating wavelet scale function by using iterative interpolation method is presented, and the derivative of scale function is used in spatial differentiation of discrete Maxwell equations. The forward modeling formula of GPR based on the interpolation wavelet scale method is derived by

  10. The extrapolated explicit midpoint scheme for variable order and step size controlled integration of the Landau-Lifschitz-Gilbert equation

    Exl, Lukas; Mauser, Norbert J.; Schrefl, Thomas; Suess, Dieter


    A practical and efficient scheme for the higher order integration of the Landau-Lifschitz-Gilbert (LLG) equation is presented. The method is based on extrapolation of the two-step explicit midpoint rule and incorporates adaptive time step and order selection. We make use of a piecewise time-linear stray field approximation to reduce the necessary work per time step. The approximation to the interpolated operator is embedded into the extrapolation process to keep in step with the hierarchic order structure of the scheme. We verify the approach by means of numerical experiments on a standardized NIST problem and compare with a higher order embedded Runge-Kutta formula. The efficiency of the presented approach increases when the stray field computation takes a larger portion of the costs for the effective field evaluation.

  11. An efficient conservative cut-cell method for rigid bodies interacting with viscous compressible flows

    Schneiders, Lennart; Günther, Claudia; Meinke, Matthias; Schröder, Wolfgang


    A Cartesian cut-cell method for viscous flows interacting with freely moving boundaries is presented. The method enables a sharp resolution of the embedded boundaries and strictly conserves mass, momentum, and energy. A new explicit Runge-Kutta scheme (PC-RK) is introduced by which the overall computational time is reduced by a factor of up to 2.5. The new scheme is a predictor-corrector type reformulation of a popular class of Runge-Kutta methods which substantially reduces the computational effort for tracking the moving boundaries and subsequently reinitializing the solver impairing neither stability nor accuracy. The structural motion is computed by an implicit scheme with good stability properties due to a strong-coupling strategy and the conservative discretization of the flow solver at the material interfaces. A new formulation for the treatment of small cut cells is proposed with high accuracy and robustness for arbitrary geometries based on a weighted Taylor-series approach solved via singular-value decomposition. The efficiency and the accuracy of the new method are demonstrated for several three-dimensional cases of laminar and turbulent particulate flow. It is shown that the new method remains fully conservative even for large displacements of the boundaries leading to a fast convergence of the fluid-solid coupling while spurious force oscillations inherent to this class of methods are effectively suppressed. The results substantiate the good stability and accuracy properties of the scheme even on relatively coarse meshes.

  12. Propagation of internal errors in explicit Runge–Kutta methods and internal stability of SSP and extrapolation methods

    Ketcheson, David I.


    In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.

  13. Resolução Numérica da equação diferencial de resfriamento de newton pelos métodos de EULER e RUNGE-KUTTA

    Andresa Pescador


    Full Text Available O presente artigo apresenta as equações diferenciais de primeira ordem, as quais constituem um ramo muito importante da matemática, pois têm uma grande aplicabilidade, tanto na matemática, como na física, biologia e também na economia. O objetivo deste estudo foi analisar a resolução de uma equação diferencial de primeira ordem, em especial a equação que define a lei de resfriamento de Newton. Verificar seu comportamento utilizando algumas aplicações, que podem ser utilizadas em sala de aula como instrumento de auxílio ao professor na abordagem destes conteúdos trazendo respostas aos questionamentos dos estudantes e motivando-os na construção de seu conhecimento. Para a resolução de uma das aplicações apresentadas buscou-se como complemento sua resolução através de dois métodos numéricos, método de Euler e método de Runge-Kutta. E por fim, fez-se uma comparação da aproximação da solução dada pela resolução numérica com a resolução analítica cuja solução é exata.

  14. Multi-Symplectic Runge-Kutta Methods for Landau-Ginzburg-Higgs Equation%Landau-Ginzburg-Higgs方程的多辛Runge-Kutta方法

    胡伟鹏; 邓子辰; 韩松梅; 范玮



  15. Dynamical Monte Carlo method for stochastic epidemic models

    Aiello, O E


    A new approach to Dynamical Monte Carlo Methods is introduced to simulate markovian processes. We apply this approach to formulate and study an epidemic Generalized SIRS model. The results are in excellent agreement with the forth order Runge-Kutta method in a region of deterministic solution. Introducing local stochastic interactions, the Runge-Kutta method is not applicable, and we solve and check it self-consistently with a stochastic version of the Euler Method. The results are also analyzed under the herd-immunity concept.

  16. Robust Numerical Methods for Nonlinear Wave-Structure Interaction in a Moving Frame of Reference

    Kontos, Stavros; Lindberg, Ole

    indicator that performs as well as the tabulated versions is proposed. Explicit high-order Runge-Kutta time integration and a Lax-Friedrichs-type numerical flux complete the scheme. The solver was tested on the two-dimensional zero speed wave radiation problem and the steady forward speed problem...

  17. Comparison of boundedness and monotonicity properties of one-leg and linear multistep methods

    Mozartova, A.


    © 2014 Elsevier B.V. All rights reserved. One-leg multistep methods have some advantage over linear multistep methods with respect to storage of the past results. In this paper boundedness and monotonicity properties with arbitrary (semi-)norms or convex functionals are analyzed for such multistep methods. The maximal stepsize coefficient for boundedness and monotonicity of a one-leg method is the same as for the associated linear multistep method when arbitrary starting values are considered. It will be shown, however, that combinations of one-leg methods and Runge-Kutta starting procedures may give very different stepsize coefficients for monotonicity than the linear multistep methods with the same starting procedures. Detailed results are presented for explicit two-step methods.

  18. Krylov subspace methods for the solution of large systems of ODE's

    Thomsen, Per Grove; Bjurstrøm, Nils Henrik


    In Air Pollution Modelling large systems of ODE's arise. Solving such systems may be done efficientliy by Semi Implicit Runge-Kutta methods. The internal stages may be solved using Krylov subspace methods. The efficiency of this approach is investigated and verified.......In Air Pollution Modelling large systems of ODE's arise. Solving such systems may be done efficientliy by Semi Implicit Runge-Kutta methods. The internal stages may be solved using Krylov subspace methods. The efficiency of this approach is investigated and verified....

  19. Extended generalized Lagrangian multipliers for magnetohydrodynamics using adaptive multiresolution methods

    Domingues M. O.


    Full Text Available We present a new adaptive multiresoltion method for the numerical simulation of ideal magnetohydrodynamics. The governing equations, i.e., the compressible Euler equations coupled with the Maxwell equations are discretized using a finite volume scheme on a two-dimensional Cartesian mesh. Adaptivity in space is obtained via Harten’s cell average multiresolution analysis, which allows the reliable introduction of a locally refined mesh while controlling the error. The explicit time discretization uses a compact Runge–Kutta method for local time stepping and an embedded Runge-Kutta scheme for automatic time step control. An extended generalized Lagrangian multiplier approach with the mixed hyperbolic-parabolic correction type is used to control the incompressibility of the magnetic field. Applications to a two-dimensional problem illustrate the properties of the method. Memory savings and numerical divergences of magnetic field are reported and the accuracy of the adaptive computations is assessed by comparing with the available exact solution.

  20. Solution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method

    H. M. Abdelhafez


    Full Text Available The modified differential transform method (MDTM, Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obtained using the fourth-order Runge-Kutta numerical solution method. A comparison of the result by the numerical Runge-Kutta fourth-order accuracy method is compared with the result by the MDTM and plotted in a long time domain.

  1. Convergence analysis of combinations of different methods

    Kang, Y. [Clarkson Univ., Potsdam, NY (United States)


    This paper provides a convergence analysis for combinations of different numerical methods for solving systems of differential equations. The author proves that combinations of two convergent linear multistep methods or Runge-Kutta methods produce a new convergent method of which the order is equal to the smaller order of the two original methods.

  2. Solution of Constrained Optimal Control Problems Using Multiple Shooting and ESDIRK Methods

    Capolei, Andrea; Jørgensen, John Bagterp


    In this paper, we describe a novel numerical algorithm for solution of constrained optimal control problems of the Bolza type for stiff and/or unstable systems. The numerical algorithm combines explicit singly diagonally implicit Runge-Kutta (ESDIRK) integration methods with a multiple shooting...... algorithm. As we consider stiff systems, implicit solvers with sensitivity computation capabilities for initial value problems must be used in the multiple shooting algorithm. Traditionally, multi-step methods based on the BDF algorithm have been used for such problems. The main novel contribution...... of this paper is the use of ESDIRK integration methods for solution of the initial value problems and the corresponding sensitivity equations arising in the multiple shooting algorithm. Compared to BDF-methods, ESDIRK-methods are advantageous in multiple shooting algorithms in which restarts and frequent...

  3. A comparison of numerical methods for non-Newtonian fluid flows in a sudden expansion

    Ilio, G. Di; Chiappini, D.; Bella, G.


    A numerical study on incompressible laminar flow in symmetric channel with sudden expansion is conducted. In this work, Newtonian and non-Newtonian fluids are considered, where non-Newtonian fluids are described by the power-law model. Three different computational methods are employed, namely a semi-implicit Chorin projection method (SICPM), an explicit algorithm based on fourth-order Runge-Kutta method (ERKM) and a Lattice Boltzmann method (LBM). The aim of the work is to investigate on the capabilities of the LBM for the solution of complex flows through the comparison with traditional computational methods. In the range of Reynolds number investigated, excellent agreement with the literature results is found. In particular, the LBM is found to be accurate in the prediction of the fluid flow behavior for the problem under consideration.

  4. Method of moving frames to solve the shallow water equations on arbitrary rotating curved surfaces

    Chun, S.; Eskilsson, C.


    A novel numerical scheme is proposed to solve the shallow water equations (SWEs) on arbitrary rotating curved surfaces. Based on the method of moving frames (MMF) in which the geometry is represented by orthonormal vectors, the proposed scheme not only has the fewest dimensionality both in space and time, but also does not require either of metric tensors, composite meshes, or the ambient space. The MMF-SWE formulation is numerically discretized using the discontinuous Galerkin method of arbitrary polynomial order p in space and an explicit Runge-Kutta scheme in time. The numerical model is validated against six standard tests on the sphere and the optimal order of convergence of p + 1 is numerically demonstrated. The MMF-SWE scheme is also demonstrated for its efficiency and stability on the general rotating surfaces such as ellipsoid, irregular, and non-convex surfaces.

  5. The 3D Lagrangian Integral Method. Henrik Koblitz Rasmussen

    Rasmussen, Henrik Koblitz


    with a second order Runge-Kutta integration method. In any development of a numerical method for viscoelastic flow it is important to focus on the constitutive equation associated to the method. For instance the K-BKZ model is not adequate to describe both shear and extensional flow using the same constitutive...

  6. A parallel method for numerical solution of delay differential equations


    A parallel diagonally-iterated Runge-Kutta (PDIRK) method is constructed to solve stiff initial value problems for delay differential equations. The order and stability of this PDIRK method has been analyzed, and the iteration parameters of the method are tuned in such a way that fast convergence to the value of corrector is achieved.

  7. Numerical methods for the stochastic Landau-Lifshitz Navier-Stokes equations.

    Bell, John B; Garcia, Alejandro L; Williams, Sarah A


    The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several computational fluid dynamics approaches are considered (including MacCormack's two-step Lax-Wendroff scheme and the piecewise parabolic method) and are found to give good results for the variance of momentum fluctuations. However, neither of these schemes accurately reproduces the fluctuations in energy or density. We introduce a conservative centered scheme with a third-order Runge-Kutta temporal integrator that does accurately produce fluctuations in density, energy, and momentum. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS solver are compared with theory, when available, and with molecular simulations using a direct simulation Monte Carlo algorithm.

  8. Energy conservation in explicit time integrators for the Navier-Stokes equations

    Capuano, Francesco; Coppola, Gennaro; Luis, Rández; Luigi, De Luca


    Discrete conservation of kinetic energy is a fundamental requirement in the numerical solution of the incompressible Navier-Stokes equations. A fully conservative algorithm requires that both the spatial and temporal discretizations do not spuriously contribute to the discrete global energy balance. While various methods are available to accomplish spatial conservation, algorithms that preserve kinetic energy exactly in time are necessarily implicit and might be not applicable in practical situations. In this work, explicit Runge-Kutta methods with optimal energy conservation properties are investigated. The proposed methods are designed to be accurate to order p and to preserve kinetic energy to order q, with q > p . The beneficial effects of the proposed methods have been assessed in terms of a properly defined effective Reynolds number, taking into account both numerical and physical viscosity. Numerical simulations of the three-dimensional Taylor-Green Vortex at high Reynolds number have shown that the proposed methods are able to keep the effective Reynolds number of the flow very close to the nominal one, while classical explicit schemes show large discrepancies.

  9. Cheap arbitrary high order methods for single integrand SDEs

    Debrabant, Kristian; Kværnø, Anne


    For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order $p_d$ we obtain methods converging in the mean-square and weak sense with order $\\lfloor p_d/2\\rfloor$. The reason is that the B...

  10. High—Order Gas—Kinetic Methods for Ideal Magnetohydrodynamics



    This article is to study extension of gas-kinetic theory based flux splitting methods to ideal magnetohydrodynamics(MHD) equations,Uniform high-order gas-kinetic methods are presented,based on TVD type RUnge-Kutta time discretization and technique of the initial reconstruction.The numerical results have been given to show robustness of our schemes.

  11. High-order weighted essentially nonoscillatory finite-difference formulation of the lattice Boltzmann method in generalized curvilinear coordinates

    Hejranfar, Kazem; Saadat, Mohammad Hossein; Taheri, Sina


    In this work, a high-order weighted essentially nonoscillatory (WENO) finite-difference lattice Boltzmann method (WENOLBM) is developed and assessed for an accurate simulation of incompressible flows. To handle curved geometries with nonuniform grids, the incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation is transformed into the generalized curvilinear coordinates and the spatial derivatives of the resulting lattice Boltzmann equation in the computational plane are solved using the fifth-order WENO scheme. The first-order implicit-explicit Runge-Kutta scheme and also the fourth-order Runge-Kutta explicit time integrating scheme are adopted for the discretization of the temporal term. To examine the accuracy and performance of the present solution procedure based on the WENOLBM developed, different benchmark test cases are simulated as follows: unsteady Taylor-Green vortex, unsteady doubly periodic shear layer flow, steady flow in a two-dimensional (2D) cavity, steady cylindrical Couette flow, steady flow over a 2D circular cylinder, and steady and unsteady flows over a NACA0012 hydrofoil at different flow conditions. Results of the present solution are compared with the existing numerical and experimental results which show good agreement. To show the efficiency and accuracy of the solution methodology, the results are also compared with the developed second-order central-difference finite-volume lattice Boltzmann method and the compact finite-difference lattice Boltzmann method. It is shown that the present numerical scheme is robust, efficient, and accurate for solving steady and unsteady incompressible flows even at high Reynolds number flows.

  12. High-order weighted essentially nonoscillatory finite-difference formulation of the lattice Boltzmann method in generalized curvilinear coordinates.

    Hejranfar, Kazem; Saadat, Mohammad Hossein; Taheri, Sina


    In this work, a high-order weighted essentially nonoscillatory (WENO) finite-difference lattice Boltzmann method (WENOLBM) is developed and assessed for an accurate simulation of incompressible flows. To handle curved geometries with nonuniform grids, the incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation is transformed into the generalized curvilinear coordinates and the spatial derivatives of the resulting lattice Boltzmann equation in the computational plane are solved using the fifth-order WENO scheme. The first-order implicit-explicit Runge-Kutta scheme and also the fourth-order Runge-Kutta explicit time integrating scheme are adopted for the discretization of the temporal term. To examine the accuracy and performance of the present solution procedure based on the WENOLBM developed, different benchmark test cases are simulated as follows: unsteady Taylor-Green vortex, unsteady doubly periodic shear layer flow, steady flow in a two-dimensional (2D) cavity, steady cylindrical Couette flow, steady flow over a 2D circular cylinder, and steady and unsteady flows over a NACA0012 hydrofoil at different flow conditions. Results of the present solution are compared with the existing numerical and experimental results which show good agreement. To show the efficiency and accuracy of the solution methodology, the results are also compared with the developed second-order central-difference finite-volume lattice Boltzmann method and the compact finite-difference lattice Boltzmann method. It is shown that the present numerical scheme is robust, efficient, and accurate for solving steady and unsteady incompressible flows even at high Reynolds number flows.

  13. Development of an explicit non-staggered scheme for solving three-dimensional Maxwell's equations

    Sheu, Tony W. H.; Chung, Y. W.; Li, J. H.; Wang, Y. C.


    An explicit finite-difference scheme for solving the three-dimensional Maxwell's equations in non-staggered grids is presented. We aspire to obtain time-dependent solutions of the Faraday's and Ampère's equations and predict the electric and magnetic fields within the discrete zero-divergence context (or Gauss's law). The local conservation laws in Maxwell's equations are numerically preserved using the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme. Following the method of lines, the spatial derivative terms in the semi-discretized Faraday's and Ampère's equations are approximated theoretically to obtain a highly accurate numerical phase velocity. The proposed fourth-order accurate space-centered finite difference scheme minimizes the discrepancy between the exact and numerical phase velocities. This minimization process considerably reduces the dispersion and anisotropy errors normally associated with finite difference time-domain methods. The computational efficiency of getting the same level of accuracy at less computing time and the ability of preserving the symplectic property have been numerically demonstrated through several test problems.

  14. Differential Transform Method for Mathematical Modeling of Jamming Transition Problem in Traffic Congestion Flow

    Ganji, S. S.; Barari, Amin; Ibsen, Lars Bo


    . In current research the authors utilized the Differential Transformation Method (DTM) for solving the nonlinear problem and compared the analytical results with those ones obtained by the 4th order Runge-Kutta Method (RK4) as a numerical method. Further illustration embedded in this paper shows the ability...

  15. Differential Transform Method for Mathematical Modeling of Jamming Transition Problem in Traffic Congestion Flow

    Ganji, S.; Barari, Amin; Ibsen, Lars Bo


    . In current research the authors utilized the Differential Transformation Method (DTM) for solving the nonlinear problem and compared the analytical results with those ones obtained by the 4th order Runge-Kutta Method (RK4) as a numerical method. Further illustration embedded in this paper shows the ability...

  16. Application of homotopy-perturbation method to nonlinear population dynamics models

    Chowdhury, M.S.H. [School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor (Malaysia); Hashim, I. [School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor (Malaysia)], E-mail:; Abdulaziz, O. [School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor (Malaysia)


    In this Letter, the homotopy-perturbation method (HPM) is employed to derive approximate series solutions of nonlinear population dynamics models. The nonlinear models considered are the multispecies Lotka-Volterra equations. The accuracy of this method is examined by comparison with the available exact and the fourth-order Runge-Kutta method (RK4)

  17. Parallel Störmer-Cowell methods for high-precision orbit computations

    Houwen, P.J. van der; Messina, E.; Swart, J.J.B. de


    Many orbit problems in celestial mechanics are described by (nonstiff) initial-value problems (IVPs) for second-order ordinary differential equations of the form $y' = {bf f (y)$. The most successful integration methods are based on high-order Runge-Kutta-Nyström formulas. However, these methods wer

  18. A Class of Optimal-Order Zero-Finding Methods Using Derivative Evaluations


    order, x1 = x(tn + h) + 0(h ) , so our nonlinear Runge-Kutta method has order three by the usual definition of order ( Henrici [62]). Similarly...mmmmmmim ll!,IW!.lnl|ll,|HMIIl.pilwmpiW, . 14, Henrici 162] Henrici , P., "Discrete variable methods in or- dinary differential equations", Wiley


    Ai-guo Xiao


    The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissipative dynamical systems, and to extend the existing relevant results of Runge-Kutta methods by Humphries and Stuart(1994).

  20. Report of the Fifth Biennial Conference on Chemical Education. Birds-of-a-Feather Workshop Sessions: Numerical Methods Workshop.

    Johnson, K. J.


    This workshop was for participants who were interested in developing a numerical methods course. The contents of a numerical methods text were covered, with special emphasis on nonlinear least squares analysis, and the Runge-Kutta method of integrating systems of first-order differential equations. (BB)

  1. Advancing parabolic operators in thermodynamic MHD models: Explicit super time-stepping versus implicit schemes with Krylov solvers

    Caplan, R. M.; Mikić, Z.; Linker, J. A.; Lionello, R.


    We explore the performance and advantages/disadvantages of using unconditionally stable explicit super time-stepping (STS) algorithms versus implicit schemes with Krylov solvers for integrating parabolic operators in thermodynamic MHD models of the solar corona. Specifically, we compare the second-order Runge-Kutta Legendre (RKL2) STS method with the implicit backward Euler scheme computed using the preconditioned conjugate gradient (PCG) solver with both a point-Jacobi and a non-overlapping domain decomposition ILU0 preconditioner. The algorithms are used to integrate anisotropic Spitzer thermal conduction and artificial kinematic viscosity at time-steps much larger than classic explicit stability criteria allow. A key component of the comparison is the use of an established MHD model (MAS) to compute a real-world simulation on a large HPC cluster. Special attention is placed on the parallel scaling of the algorithms. It is shown that, for a specific problem and model, the RKL2 method is comparable or surpasses the implicit method with PCG solvers in performance and scaling, but suffers from some accuracy limitations. These limitations, and the applicability of RKL methods are briefly discussed.

  2. On advanced variational formulation of the method of lines and its application to the wave propagation problems

    Shatalov, M


    Full Text Available of elastic wave over the bar and . If we add initial conditions and to the system of ordinary differential equations (3.5) the initial problem will be formulated and could be solved by one of the available numerical methods (Runge-Kutta, Adams, etc...

  3. On symplectic and symmetric ARKN methods

    Shi, Wei; Wu, Xinyuan


    Symplecticness and symmetry are favorable properties for solving Hamiltonian systems. For the oscillatory second-order initial value problems of the form q+ωq=f(q,q), adapted Runge-Kutta-Nyström methods (ARKN methods, in short notation) were investigated by several authors. In a wide range of physical applications from molecular dynamics to nonlinear wave propagation, an important class of the problems is Hamiltonian systems for which symplectic methods should be preferred. Hence it is quite natural to raise a question of the symplecticness for ARKN methods. In this paper we investigate the symplecticness conditions of ARKN methods for separable Hamiltonian systems. We conclude that there exist only one-stage explicit symplectic ARKN (SARKN, in short notation) methods under the symplecticness conditions of ARKN methods. The SARKN methods have a special form and the algebraic order cannot exceed 2. We also point out that no ARKN method can be symmetric. An explicit SARKN method of order two is proposed with the analysis of phase and stability properties. The numerical results accompanied show good performance for the new explicit symplectic algorithm in comparison with the popular symplectic methods in the scientific literature.


    Hong-yu Liu; Geng Sun


    Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned RungeKutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can't reach order more than s + 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s + 1 when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.


    Ai-guo Xiao; Shou-fu Li; Min Yang


    In this paper, we present some invariants and conservation laws of general linear methods applied to differential equation systems. We show that the quadratic invariants and symplecticity of the systems can be extended to general linear methods by a tensor product, and show that general linear methods with the matrix M=0 inherit in an extended sense the quadratic invariants possessed by the differential equation systems being integrated and preserve in an extended sense the symplectic structure of the phase space in the integration of Hamiltonian systems. These unify and extend existing relevant results on Runge-Kutta methods, linear multistep methods and one-leg methods. Finally, as special cases of general linear methods, we examine multistep Runge-Kutta methods, one-leg methods and linear two-step methods in detail.

  6. The Full Implicit Runge-Kutta Method of Solving Point Reactor Kinetics Equations%全隐式龙格库塔法求解点堆动力学方程

    王伟吉; 叶金亮; 方成跃



  7. Fourth order difference methods for hyperbolic IBVP's

    Gustafsson, Bertil; Olsson, Pelle


    Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

  8. Workshop on Numerical Methods for Ordinary Differential Equations

    Gear, Charles; Russo, Elvira


    Developments in numerical initial value ode methods were the focal topic of the meeting at L'Aquila which explord the connections between the classical background and new research areas such as differental-algebraic equations, delay integral and integro-differential equations, stability properties, continuous extensions (interpolants for Runge-Kutta methods and their applications, effective stepsize control, parallel algorithms for small- and large-scale parallel architectures). The resulting proceedings address many of these topics in both research and survey papers.

  9. A Comparison Study of the Eigenvalue Method for the Solution of the Transient Heat Conduction Equation.


    1966). 3. Canale, R.P. and S.C. Chapra . Numerical Methods for Engineers with Personnel Computer Applications. New York: McGraw-Hill 509-533, ( 1985...This study looks at numerical % methods from an engineer’s view, a tool to be used in solving problems. This paper has given me much needed experience... numerical method in solving the transient heat conduction equation. The eigenvalue method was compared to five other numerical methods : Runge-Kutta

  10. A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations

    Hu, Changqing; Shu, Chi-Wang


    In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.

  11. Numerical Simulation of Coupled Nonlinear Schr(o)dinger Equations Using the Generalized Differential Quadrature Method

    R.Mokhtari; A.Samadi Toodar; N.G.Chegini


    @@ We the extend application of the generalized differential quadrature method (GDQM) to solve some coupled nonlinear Schr(o)dinger equations.The cosine-based GDQM is employed and the obtained system of ordinary differential equations is solved via the fourth order Runge-Kutta method.The numerical solutions coincide with the exact solutions in desired machine precision and invariant quantities are conserved sensibly.Some comparisons with the methods applied in the literature are carried out.%We the extend application of the generalized differential quadrature method (GDQM) to solve some coupled nonlinear Schrodinger equations. The cosine-based GDQM is employed and the obtained system of ordinary differential equations is solved via the fourth order Runge-Kutta method. The numerical solutions coincide with the exact solutions in desired machine precision and invariant quantities are conserved sensibly. Some comparisons with the methods applied in the literature are carried out.

  12. Fourth-Order Difference Methods for Hyperbolic IBVPs

    Gustafsson, Bertil; Olsson, Pelle


    In this paper we consider fourth-order difference approximations of initial-boundary value problems for hyperbolic partial differential equations. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics; the second one is used for modeling shocks and rarefaction waves. The time discretization is done with a third-order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second-order viscosity. In case of the non-linear Burgers' equation we use a flux splitting technique that results in an energy estimate for certain difference approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth-order methods with a standard second-order one and with a third-order TVD method. The results show that the fourth-order methods are the only ones that give good results for all the considered test problems.

  13. A multistage differential transformation method for approximate solution of Hantavirus infection model

    Gökdoğan, Ahmet; Merdan, Mehmet; Yildirim, Ahmet


    The goal of this study is presented a reliable algorithm based on the standard differential transformation method (DTM), which is called the multi-stage differential transformation method (MsDTM) for solving Hantavirus infection model. The results obtanied by using MsDTM are compared to those obtained by using the Runge-Kutta method (R-K-method). The proposed technique is a hopeful tool to solving for a long time intervals in this kind of systems.

  14. Convergence analysis for general linear methods applied to stiff delay differential equations


    For Runge-Kutta methods applied to stiff delay differential equations (DDEs), the concept of D-convergence was proposed, which is an extension to that of B-convergence in ordinary differential equations (ODEs). In this paper, D-convergence of general linear methods is discussed and the previous related results are improved. Some order results to determine D-convergence of the methods are obtained.

  15. Application of Piecewise Successive Linearization Method for the Solutions of the Chen Chaotic System

    S. S. Motsa


    Full Text Available This paper centres on the application of the new piecewise successive linearization method (PSLM in solving the chaotic and nonchaotic Chen system. Numerical simulations are presented graphically and comparison is made between the PSLM and Runge-Kutta-based methods. The work shows that the proposed method provides good accuracy and can be easily extended to other dynamical systems including those that are chaotic in nature.


    Si-qing Gan; Geng Sun


    In this paper we analyze the error behavior of general linear methods applied to some classes of one-parameter multiply stiff singularly perturbed problems. We obtain the global error estimate of algebraically and diagonally stable general linear methods. The main result of this paper can be viewed as an extension of that obtained by Xiao [13] for the case of Runge-Kutta methods.

  17. Series Solution of the Multispecies Lotka-Volterra Equations by Means of the Homotopy Analysis Method

    A. Sami Bataineh


    Full Text Available The time evolution of the multispecies Lotka-Volterra system is investigated by the homotopy analysis method (HAM. The continuous solution for the nonlinear system is given, which provides a convenient and straightforward approach to calculate the dynamics of the system. The HAM continuous solution generated by polynomial base functions is of comparable accuracy to the purely numerical fourth-order Runge-Kutta method. The convergence theorem for the three-dimensional case is also given.

  18. A discontinous Galerkin finite element method with an efficient time integration scheme for accurate simulations

    Liu, Meilin


    A discontinuous Galerkin finite element method (DG-FEM) with a highly-accurate time integration scheme is presented. The scheme achieves its high accuracy using numerically constructed predictor-corrector integration coefficients. Numerical results show that this new time integration scheme uses considerably larger time steps than the fourth-order Runge-Kutta method when combined with a DG-FEM using higher-order spatial discretization/basis functions for high accuracy. © 2011 IEEE.

  19. General linear methods and friends: Toward efficient solutions of multiphysics problems

    Sandu, Adrian


    Time dependent multiphysics partial differential equations are of great practical importance as they model diverse phenomena that appear in mechanical and chemical engineering, aeronautics, astrophysics, meteorology and oceanography, financial modeling, environmental sciences, etc. There is no single best time discretization for the complex multiphysics systems of practical interest. We discuss "multimethod" approaches that combine different time steps and discretizations using the rigourous frameworks provided by Partitioned General Linear Methods and Generalize-structure Additive Runge Kutta Methods..

  20. A Review of High-Order and Optimized Finite-Difference Methods for Simulating Linear Wave Phenomena

    Zingg, David W.


    This paper presents a review of high-order and optimized finite-difference methods for numerically simulating the propagation and scattering of linear waves, such as electromagnetic, acoustic, or elastic waves. The spatial operators reviewed include compact schemes, non-compact schemes, schemes on staggered grids, and schemes which are optimized to produce specific characteristics. The time-marching methods discussed include Runge-Kutta methods, Adams-Bashforth methods, and the leapfrog method. In addition, the following fourth-order fully-discrete finite-difference methods are considered: a one-step implicit scheme with a three-point spatial stencil, a one-step explicit scheme with a five-point spatial stencil, and a two-step explicit scheme with a five-point spatial stencil. For each method studied, the number of grid points per wavelength required for accurate simulation of wave propagation over large distances is presented. Recommendations are made with respect to the suitability of the methods for specific problems and practical aspects of their use, such as appropriate Courant numbers and grid densities. Avenues for future research are suggested.

  1. Accelerating finite-rate chemical kinetics with coprocessors: comparing vectorization methods on GPUs, MICs, and CPUs

    Stone, Christopher P


    Efficient ordinary differential equation solvers for chemical kinetics must take into account the available thread and instruction-level parallelism of the underlying hardware, especially on many-core coprocessors, as well as the numerical efficiency. A stiff Rosenbrock and nonstiff Runge-Kutta solver are implemented using the single instruction, multiple thread (SIMT) and single instruction, multiple data (SIMD) paradigms with OpenCL. The performances of these parallel implementations were measured with three chemical kinetic models across several multicore and many-core platforms. Two runtime benchmarks were conducted to clearly determine any performance advantage offered by either method: evaluating the right-hand-side source terms in parallel, and integrating a series of constant-pressure homogeneous reactors using the Rosenbrock and Runge-Kutta solvers. The right-hand-side evaluations with SIMD parallelism on the host multicore Xeon CPU and many-core Xeon Phi co-processor performed approximately three ti...

  2. Direct SQP-methods for solving optimal control problems with delays

    Goellmann, L.; Bueskens, C.; Maurer, H.


    The maximum principle for optimal control problems with delays leads to a boundary value problem (BVP) which is retarded in the state and advanced in the costate function. Based on shooting techniques, solution methods for this type of BVP have been proposed. In recent years, direct optimization methods have been favored for solving control problems without delays. Direct methods approximate the control and the state over a fixed mesh and solve the resulting NLP-problem with SQP-methods. These methods dispense with the costate function and have shown to be robust and efficient. In this paper, we propose a direct SQP-method for retarded control problems. In contrast to conventional direct methods, only the control variable is approximated by e.g. spline-functions. The state is computed via a high order Runge-Kutta type algorithm and does not enter explicitly the NLP-problem through an equation. This approach reduces the number of optimization variables considerably and is implementable even on a PC. Our method is illustrated by the numerical solution of retarded control problems with constraints. In particular, we consider the control of a continuous stirred tank reactor which has been solved by dynamic programming. This example illustrates the robustness and efficiency of the proposed method. Open questions concerning sufficient conditions and convergence of discretized NLP-problems are discussed.

  3. A Newton-Krylov method with an approximate analytical Jacobian for implicit solution of Navier-Stokes equations on staggered overset-curvilinear grids with immersed boundaries

    Asgharzadeh, Hafez; Borazjani, Iman


    The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for non-linear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form a preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the

  4. Comparison of Several Numerical Methods for Simulation of Compressible Shear Layers

    Kennedy, Christopher A.; Carpenter, Mark H.


    An investigation is conducted on several numerical schemes for use in the computation of two-dimensional, spatially evolving, laminar variable-density compressible shear layers. Schemes with various temporal accuracies and arbitrary spatial accuracy for both inviscid and viscous terms are presented and analyzed. All integration schemes use explicit or compact finite-difference derivative operators. Three classes of schemes are considered: an extension of MacCormack's original second-order temporally accurate method, a new third-order variant of the schemes proposed by Rusanov and by Kutier, Lomax, and Warming (RKLW), and third- and fourth-order Runge-Kutta schemes. In each scheme, stability and formal accuracy are considered for the interior operators on the convection-diffusion equation U(sub t) + aU(sub x) = alpha U(sub xx). Accuracy is also verified on the nonlinear problem, U(sub t) + F(sub x) = 0. Numerical treatments of various orders of accuracy are chosen and evaluated for asymptotic stability. Formally accurate boundary conditions are derived for several sixth- and eighth-order central-difference schemes. Damping of high wave-number data is accomplished with explicit filters of arbitrary order. Several schemes are used to compute variable-density compressible shear layers, where regions of large gradients exist.

  5. Spectral Element Method for the Simulation of Unsteady Compressible Flows

    Diosady, Laslo Tibor; Murman, Scott M.


    This work uses a discontinuous-Galerkin spectral-element method (DGSEM) to solve the compressible Navier-Stokes equations [1{3]. The inviscid ux is computed using the approximate Riemann solver of Roe [4]. The viscous fluxes are computed using the second form of Bassi and Rebay (BR2) [5] in a manner consistent with the spectral-element approximation. The method of lines with the classical 4th-order explicit Runge-Kutta scheme is used for time integration. Results for polynomial orders up to p = 15 (16th order) are presented. The code is parallelized using the Message Passing Interface (MPI). The computations presented in this work are performed using the Sandy Bridge nodes of the NASA Pleiades supercomputer at NASA Ames Research Center. Each Sandy Bridge node consists of 2 eight-core Intel Xeon E5-2670 processors with a clock speed of 2.6Ghz and 2GB per core memory. On a Sandy Bridge node the Tau Benchmark [6] runs in a time of 7.6s.

  6. A meshless local radial basis function method for two-dimensional incompressible Navier-Stokes equations

    Wang, Zhiheng


    A meshless local radial basis function method is developed for two-dimensional incompressible Navier-Stokes equations. The distributed nodes used to store the variables are obtained by the philosophy of an unstructured mesh, which results in two main advantages of the method. One is that the unstructured nodes generation in the computational domain is quite simple, without much concern about the mesh quality; the other is that the localization of the obtained collocations for the discretization of equations is performed conveniently with the supporting nodes. The algebraic system is solved by a semi-implicit pseudo-time method, in which the convective and source terms are explicitly marched by the Runge-Kutta method, and the diffusive terms are implicitly solved. The proposed method is validated by several benchmark problems, including natural convection in a square cavity, the lid-driven cavity flow, and the natural convection in a square cavity containing a circular cylinder, and very good agreement with the existing results are obtained.

  7. High-Order Space-Time Methods for Conservation Laws

    Huynh, H. T.


    Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown

  8. Vibratory gyroscopes : identification of mathematical model from test data

    Shatalov, MY


    Full Text Available by an adaptive Runge-Kutta method with the same initial conditions as from the test data. Results of comparison of the test data and results of numerical integration of equations (6) are shown in Fig.3-6. Fig. 3. Runge-Kutta and test data of in-phase X...-channel components (---- - Runge-Kutta integration; - - - - test data) Fig. 4. Runge-Kutta and test data of quadrature X-channel components (---- - Runge-Kutta integration; - - - - test data) Fig. 5. Runge-Kutta and test data of in-phase Y...

  9. Level Set interface treatment and its application in Euler method


    Level Set interface treatment method is introduced into Euler method,which is employed for interface treatment method for multi-materials. Combined with the ghost fluid method,the moving interface is tracked. Fifth-order WENO spatial discretization and third-order TVD Runge-Kutta time discretization methods are used. Shock-wave action on bubble,implosion and velocity field Shock effect bubbles; implosion and velocity field are simulated by means of LS-MMIC3D programmed by C++. Nu-merical results show that the Level Set interface treatment method is effective and feasible for multi-material interface treatment in comparison with the WENO method.

  10. Revising Geology Labs To Explicitly Use the Scientific Method.

    Hannula, Kimberly A.


    Proposes that content- or skill-based labs can be revised to explicitly involve the scientific method by asking students to propose hypotheses before making observations. Students' self-assessment showed they felt that they learned a great deal from this style of labs and found the labs to be fun; however, students felt that they learned little…

  11. Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

    Hundsdorfer, W.


    In this paper nonlinear monotonicity and boundedness properties are analyzed for linear multistep methods. We focus on methods which satisfy a weaker boundedness condition than strict monotonicity for arbitrary starting values. In this way, many linear multistep methods of practical interest are included in the theory. Moreover, it will be shown that for such methods monotonicity can still be valid with suitable Runge-Kutta starting procedures. Restrictions on the stepsizes are derived that are not only sufficient but also necessary for these boundedness and monotonicity properties. © 2011 Springer Science+Business Media, LLC.

  12. Multistage Spectral Relaxation Method for Solving the Hyperchaotic Complex Systems

    Hassan Saberi Nik


    Full Text Available We present a pseudospectral method application for solving the hyperchaotic complex systems. The proposed method, called the multistage spectral relaxation method (MSRM is based on a technique of extending Gauss-Seidel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudospectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous hyperchaotic complex systems such as hyperchaotic complex Lorenz system and the complex permanent magnet synchronous motor. We compare this approach to the Runge-Kutta based ode45 solver to show that the MSRM gives accurate results.

  13. Explicit-Explicit Sequence Calculation Method for the Wheel/rail Rolling Contact Problem Based on ANSYS/LS-DYNA

    Song Hua


    Full Text Available The wheel/rail rolling contact can not only lead to rail fatigue damage but also bring rail corrugation. According to the wheel/rail rolling contact problem, based on the ANSYS/LS-DYNA explicit analysis software, this paper established the finite element model of wheel/rail rolling contact in non-linear steady-state curve negotiation, and proposed the explicit-explicit sequence calculation method that can be used to solve this model. The explicit-explicit sequence calculation method uses explicit solver in calculating the rail pre-stressing force and the process of wheel/rail rolling contact. Compared with the implicit-explicit sequence calculation method that has been widely applied, the explicit-explicit sequence calculation method including similar precision in calculation with faster speed and higher efficiency, make it more applicable to solve the wheel/rail rolling contact problem of non-linear steady-state curving with a large solving model or a high non-linear degree.

  14. Adaptive explicit Magnus numerical method for nonlinear dynamical systems

    LI Wen-cheng; DENG Zi-chen


    Based on the new explicit Magnus expansion developed for nonlinear equations defined on a matrix Lie group,an efficient numerical method is proposed for nonlinear dynamical systems.To improve computational efficiency,the integration step size can be adaptively controlled.Validity and effectiveness of the method are shown by application to several nonlinear dynamical systems including the Duffing system,the van der Pol system with strong stiffness,and the nonlinear Hamiltonian pendulum system.

  15. On the relationship between ODE solvers and iterative solvers for linear equations

    Lorber, A.; Joubert, W.; Carey, G.F. [Univ. of Texas, Austin, TX (United States)


    The connection between the solution of linear systems of equations by both iterative methods and explicit time stepping techniques is investigated. Based on the similarities, a suite of Runge-Kutta time integration schemes with extended stability domains are developed using Chebyshev iteration polynomials. These Runge-Kutta schemes are applied to linear and non-linear systems arising from the numerical solution of PDE`s containing either physical or artificial transient terms. Specifically, the solutions of model linear convection and convection-diffusion equations are presented, as well as the solution of a representative non-linear Navier-Stokes fluid flow problem. Included are results of parallel computations.

  16. A hybrid Pseudo-spectral Immersed-Boundary Method for Applications to Aquatic Locomotion

    Ren, Zheng; Hall, David; Mohseni, Kamran


    A hybrid pseudo-spectral immersed boundary method is developed for application in marine locomotion. Spatial derivatives are calculated using pseudo-spectral method while a 2nd-order Runge-Kutta scheme is used for time integration. The singular force applied on the immersed boundary is obtained using a direct forcing method. To avoid Gibb's phenomenon in the spectral method, we regularize the force by smoothing it over several grid cells. This method has the advantage of spectral accuracy and the flexibility to model irregular, moving boundaries on a Cartesian coordinate without complex mesh generation. The method is applied to examine locomotion of jellyfish for both jetting and paddling jellyfish.

  17. An explicit high order method for fractional advection diffusion equations

    Sousa, Ercília


    We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1convergence of the numerical method through consistency and stability. The order of convergence varies between two and three and for advection dominated flows is close to three. Although the method is conditionally stable, the restrictions allow wide stability regions. The analysis is confirmed by numerical examples.

  18. An explicit four-dimensional variational data assimilation method

    QIU ChongJian; ZHANG Lei; SHAO AiMei


    A new data assimilation method called the explicit four-dimensional variational (4DVAR) method is proposed. In this method, the singular value decomposition (SVD) is used to construct the orthogonal basis vectors from a forecast ensemble in a 4D space. The basis vectors represent not only the spatial structure of the analysis variables but also the temporal evolution. After the analysis variables are expressed by a truncated expansion of the basis vectors in the 4D space, the control variables in the cost function appear explicitly, so that the adjoint model, which is used to derive the gradient of cost function with respect to the control variables, is no longer needed. The new technique significantly simplifies the data assimilation process. The advantage of the proposed method is demonstrated by several experiments using a shallow water numerical model and the results are compared with those of the conventional 4DVAR. It is shown that when the observation points are very dense, the conventional 4DVAR is better than the proposed method. However, when the observation points are sparse, the proposed method performs better. The sensitivity of the proposed method with respect to errors in the observations and the numerical model is lower than that of the conventional method.

  19. An explicit four-dimensional variational data assimilation method


    A new data assimilation method called the explicit four-dimensional variational (4DVAR) method is proposed. In this method, the singular value decomposition (SVD) is used to construct the orthogonal basis vectors from a forecast ensemble in a 4D space. The basis vectors represent not only the spatial structure of the analysis variables but also the temporal evolution. After the analysis variables are ex-pressed by a truncated expansion of the basis vectors in the 4D space, the control variables in the cost function appear explicitly, so that the adjoint model, which is used to derive the gradient of cost func-tion with respect to the control variables, is no longer needed. The new technique significantly simpli-fies the data assimilation process. The advantage of the proposed method is demonstrated by several experiments using a shallow water numerical model and the results are compared with those of the conventional 4DVAR. It is shown that when the observation points are very dense, the conventional 4DVAR is better than the proposed method. However, when the observation points are sparse, the proposed method performs better. The sensitivity of the proposed method with respect to errors in the observations and the numerical model is lower than that of the conventional method.

  20. NMPC for Oil Reservoir Production Optimization

    Völcker, Carsten; Jørgensen, John Bagterp; Thomsen, Per Grove


    this problem numerically using a single shooting sequential quadratic programming (SQP) based optimization method. Explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods are used for integration of the stiff system of differential equations describing the two-phase flow, and the adjoint method...


    D.C. Wan; G.W. Wei


    An efficient discrete singular convolution (DSC) method is introduced to the numerical solutions of incompressible Euler and Navier-Stokes equations with periodic boundary conditions. Two numerical tests of two-dimensional NavierStokes equations with periodic boundary conditions and Euler equations for doubly periodic shear layer flows are carried out by using the DSC method for spatial derivatives and fourth-order Runge-Kutta method for time advancement, respectively. The computational results show that the DSC method is efficient and robust for solving the problems of incompressible flows, and has the potential of being extended to numerically solve much broader problems in fluid dynamics.

  2. A method of weighted residuals with trigonometric basis functions for sound transmission in circular ducts

    Vo, P. T.; Eversman, W.


    The method of weighted residuals (MWR) in the form of a modified Galerkin method with trigonometric basis functions is used to compute the transmission of sound in an axisymmetric duct. The method is used to generate the axial wave number for uniform ducts. These are compared with exact solutions generated by a formal eigenvalue routine in the hard-wall case and a Runge-Kutta integration eigenvalue scheme in the soft-wall case. The method is applicable to both flow and no-flow cases.

  3. Iterative Multistep Reproducing Kernel Hilbert Space Method for Solving Strongly Nonlinear Oscillators

    Banan Maayah


    Full Text Available A new algorithm called multistep reproducing kernel Hilbert space method is represented to solve nonlinear oscillator’s models. The proposed scheme is a modification of the reproducing kernel Hilbert space method, which will increase the intervals of convergence for the series solution. The numerical results demonstrate the validity and the applicability of the new technique. A very good agreement was found between the results obtained using the presented algorithm and the Runge-Kutta method, which shows that the multistep reproducing kernel Hilbert space method is very efficient and convenient for solving nonlinear oscillator’s models.

  4. Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

    Daniel Olvera


    Full Text Available We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM that is based on the homotopy perturbation method (HPM and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameter h by following the homotopy analysis method (HAM. At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

  5. Numerical Solutions of the Multispecies Predator-Prey Model by Variational Iteration Method

    Khaled Batiha


    Full Text Available The main objective of the current work was to solve the multispecies predator-prey model. The techniques used here were called the variational iteration method (VIM and the Adomian decomposition method (ADM. The advantage of this work is twofold. Firstly, the VIM reduces the computational work. Secondly, in comparison with existing techniques, the VIM is an improvement with regard to its accuracy and rapid convergence. The VIM has the advantage of being more concise for analytical and numerical purposes. Comparisons with the exact solution and the fourth-order Runge-Kutta method (RK4 show that the VIM is a powerful method for the solution of nonlinear equations.

  6. On the Analysis of Multistep-Out-of-Grid Method for Celestial Mechanics Tasks

    Olifer, L.; Choliy, V.


    Occasionally, there is a necessity in high-accurate prediction of celestial body trajectory. The most common way to do that is to solve Kepler's equation analytically or to use Runge-Kutta or Adams integrators to solve equation of motion numerically. For low-orbit satellites, there is a critical need in accounting geopotential and another forces which influence motion. As the result, the right side of equation of motion becomes much bigger, and classical integrators will not be quite effective. On the other hand, there is a multistep-out-of-grid (MOG) method which combines Runge-Kutta and Adams methods. The MOG method is based on using m on-grid values of the solution and n × m off-grid derivative estimations. Such method could provide stable integrators of maximum possible order, O (hm+mn+n-1). The main subject of this research was to implement and analyze the MOG method for solving satellite equation of motion with taking into account Earth geopotential model (ex. EGM2008 (Pavlis at al., 2008)) and with possibility to add other perturbations such as atmospheric drag or solar radiation pressure. Simulations were made for satellites on low orbit and with various eccentricities (from 0.1 to 0.9). Results of the MOG integrator were compared with results of Runge-Kutta and Adams integrators. It was shown that the MOG method has better accuracy than classical ones of the same order and less right-hand value estimations when is working on high orders. That gives it some advantage over "classical" methods.

  7. Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II

    Bagheri, Saman; Nikkar, Ali [University of Tabriz, Tabriz (Iran, Islamic Republic of)


    This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He's variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.

  8. Free terminal time optimal control problem for the treatment of HIV infection

    Amine Hamdache


    to provide the explicit formulations of the optimal controls. The corresponding optimality system with the additional transversality condition for the terminal time is derived and solved numerically using an adapted iterative method with a Runge-Kutta fourth order scheme and a gradient method routine.

  9. Production Optimization for Two-Phase Flow in an Oil Reservoir


    settings of injection and production wells are computed by solution of a large scale constrained optimal control problem. We describe a gradient based method to compute the optimal control strategy of the water flooding process. An explicit singly diagonally implicit Runge-Kutta (ESDIRK) method...

  10. Mathematical modelling of ultrasound propagation in multi-phase flow

    Simurda, Matej


    . Acoustic media are modelled by setting the shear modulus to zero. Spatial derivatives are approximated by a Fourier collocation method allowing the use of the Fast Fourier transform while the time integration is realized by the explicit fourth order Runge-Kutta finite difference scheme. The method...

  11. Dispersion and stability analysis for a finite difference beam propagation method.

    de-Oliva-Rubio, J; Molina-Fernández, I; Godoy-Rubio, R


    Applying continuous and discrete transformation techniques, new analytical expressions to calculate dispersion and stability of a Runge- Kutta Finite Difference Beam Propagation Method (RK-FDBPM) are obtained. These expressions give immediate insight about the discretization errors introduced by the numerical method in the plane-wave spectrum domain. From these expressions a novel strategy to adequately set the mesh steps sizes of the RK-FDBPM is presented. Assessment of the method is performed by studying the propagation in several linear and nonlinear photonic devices for different spatial discretizations.

  12. Numerical Solutions of the Coupled Klein-Gordon-Schrand#246;dinger Equations by Differential Quadrature Methods

    Thoudam Roshan


    Full Text Available Numerical solutions of the coupled Klein-Gordon-Schrödinger equations is obtained by using differential quadrature methods based on polynomials and quintic B-spline functions for space discretization and Runge-Kutta fourth order for time discretization. Stability of the schemes are studied using matrix stability analysis. The accuracy and efficiency of the methods are shown by conducting some numerical experiments on test problems. The motion of single soliton and interaction of two solitons are simulated by the proposed methods.

  13. Differential Transform Method for Mathematical Modeling of Jamming Transition Problem in Traffic Congestion Flow

    Ganji, S. S.; Barari, Amin; Ibsen, Lars Bo


    In this paper we aim to find an analytical solution for jamming transition in traffic flow. Generally the Jamming Transition Problem (JTP) can be modeled via Lorentz system. So, in this way, the governing differential equation achieved is modeled in the form of a nonlinear damped oscillator....... In current research the authors utilized the Differential Transformation Method (DTM) for solving the nonlinear problem and compared the analytical results with those ones obtained by the 4th order Runge-Kutta Method (RK4) as a numerical method. Further illustration embedded in this paper shows the ability...... of DTM in solving nonlinear problems when a so accurate solution is required....

  14. Adaptive gradient-augmented level set method with multiresolution error estimation

    Kolomenskiy, Dmitry; Schneider, Kai


    A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quad-tree data structures. For adaptive time integration, an embedded Runge-Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compression with respect to the uniform grid discretization are reported.

  15. A Simple Modification of Homotopy Perturbation Method for the Solution of Blasius Equation in Semi-Infinite Domains

    M. Aghakhani


    Full Text Available A simple modification of the homotopy perturbation method is proposed for the solution of the Blasius equation with two different boundary conditions. Padé approximate is used to deal with the boundary condition at infinity. The results obtained from the analytical method are compared to Howarth’s numerical solution and fifth order Runge-Kutta Fehlberg method indicating a very good agreement. The proposed method is a simple and reliable modification of homotopy perturbation method, which does not require the existence of a small parameter, linearization of the equation, or computation of Adomian’s polynomials.

  16. Particle Methods for Geophysical Flow on the Sphere

    Bosler, Peter A.

    We present a Lagrangian Particle-Panel Method (LPPM) for geophysical fluid flow on a rotating sphere motivated by problems in atmosphere and ocean dynamics. We focus here on the barotropic vorticity equation and 2D passive scalar advection, as a step towards the development of a new dynamical core for global circulation models. The LPPM method employs the Lagrangian form of the equations of motion. The flow map is discretized as a set of Lagrangian particles and panels. Particle velocity is computed by applying a midpoint rule/point vortex approximation to the Biot-Savart integral with quadrature weights determined by the panel areas. We consider several discretizations of the sphere including the cubed sphere mesh, icosahedral triangles, and spherical Voronoi tesselations. The ordinary differential equations for particle motion are integrated by the fourth order Runge-Kutta method. Mesh distortion is addressed using a combination of adaptive mesh refinement (AMR) and a new Lagrangian remeshing procedure. In contrast with Eulerian schemes, the LPPM method avoids explicit discretization of the advective derivative. In the case of passive scalar advection, LPPM preserves tracer ranges and both linear and nonlinear tracer correlations exactly. We present results for the barotropic vorticity equation applied to several test cases including solid body rotation, Rossby-Haurwitz waves, Gaussian vortices, jet streams, and a model for the breakdown of the polar vortex during sudden stratospheric warming events. The combination of AMR and remeshing enables the LPPM scheme to efficiently resolve thin fronts and filaments that develop in the vorticity distribution. We validate the accuracy of LPPM by comparing with results obtained using the Eulerian based Lin-Rood advection scheme. We examine how energy and enstrophy conservation in the LPPM scheme are affected by the time step and spatial discretization. We conclude with a discussion of how the method may be extended to the

  17. Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes

    Brahim Benhammouda


    present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45 numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend of a perturbation parameter.

  18. Investigation on vibration of single-walled carbon nanotubes by variational iteration method

    Ahmadi Asoor, A. A.; Valipour, P.; Ghasemi, S. E.


    In this paper, the variational iteration method (VIM) has been used to investigate the non-linear vibration of single-walled carbon nanotubes (SWCNTs) based on the nonlocal Timoshenko beam theory. The accuracy of results is examined by the fourth-order Runge-Kutta numerical method. Comparison between VIM solutions with numerical results leads to highly accurate solutions. Also, the behavior of deflection and frequency in vibrations of SWCNTs are studied. The results show that frequency of single walled carbon nanotube versus amplitude increases by increasing the values of B.

  19. Explicit finite-difference lattice Boltzmann method for curvilinear coordinates.

    Guo, Zhaoli; Zhao, T S


    In this paper a finite-difference-based lattice Boltzmann method for curvilinear coordinates is proposed in order to improve the computational efficiency and numerical stability of a recent method [R. Mei and W. Shyy, J. Comput. Phys. 143, 426 (1998)] in which the collision term of the Boltzmann Bhatnagar-Gross-Krook equation for discrete velocities is treated implicitly. In the present method, the implicitness of the numerical scheme is removed by introducing a distribution function different from that being used currently. As a result, an explicit finite-difference lattice Boltzmann method for curvilinear coordinates is obtained. The scheme is applied to a two-dimensional Poiseuille flow, an unsteady Couette flow, a lid-driven cavity flow, and a steady flow around a circular cylinder. The numerical results are in good agreement with the results of previous studies. Extensions to other lattice Boltzmann models based on nonuniform meshes are also discussed.

  20. Efficient integration of stiff kinetics with phase change detection for reactive reservoir processes

    Kristensen, Morten Rode; Gerritsen, Margot G.; Thomsen, Per Grove;


    We propose the use of implicit one-step Explicit Singly Diagonal Implicit Runge-Kutta (ESDIRK) methods for integration of the stiff kinetics in reactive, compositional and thermal processes that are solved using operator-splitting type approaches. To facilitate the algorithmic development we...

  1. Nodal DG-FEM solution of high-order Boussinesq-type equations

    Engsig-Karup, Allan Peter; Hesthaven, Jan S.; Bingham, Harry B.


    functions of arbitrary order in space on each element of an unstructured computational domain. A fourth order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy...

  2. Model Predictive Control Algorithms for Pen and Pump Insulin Administration

    Boiroux, Dimitri

    (OCP) is solved using a multiple-shooting based algorithm. We use an explicit Runge-Kutta method (DOPRI45) with an adaptive stepsize for numerical integration and sensitivity computation. The OCP is solved using a Quasi-Newton sequential quadratic programming (SQP) with a linesearch and a BFGS update...

  3. Coupling Chemical Kinetics and Flashes in Reactive, Thermal and Compositional Reservoir Simulation

    Kristensen, Morten Rode; Gerritsen, Margot G.; Thomsen, Per Grove


    of convergence and error test failures by more than 50% compared to direct integration without the new algorithm. To facilitate the algorithmic development we construct a virtual kinetic cell model. We use implicit one-step ESDIRK (Explicit Singly Diagonal Implicit Runge-Kutta) methods for integration...

  4. A second-order accurate immersed boundary-lattice Boltzmann method for particle-laden flows

    Zhou, Qiang; Fan, Liang-Shih


    A new immersed boundary-lattice Boltzmann method (IB-LBM) is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The immersed boundary method (IBM) recently developed by Breugem (2012) [19] is adopted in the present method, development including the retraction technique, the multi-direct forcing method and the direct account of the inertia of the fluid contained within the particles. The present IB-LBM is, however, formulated with further improvement with the implementation of the high-order Runge-Kutta schemes in the coupled fluid-particle interaction. The major challenge to implement high-order Runge-Kutta schemes in the LBM is that the flow information such as density and velocity cannot be directly obtained at a fractional time step from the LBM since the LBM only provides the flow information at an integer time step. This challenge can be, however, overcome as given in the present IB-LBM by extrapolating the flow field around particles from the known flow field at the previous integer time step. The newly calculated fluid-particle interactions from the previous fractional time steps of the current integer time step are also accounted for in the extrapolation. The IB-LBM with high-order Runge-Kutta schemes developed in this study is validated by several benchmark applications. It is demonstrated, for the first time, that the IB-LBM has the capacity to resolve the translational and rotational motion of particles with the second-order accuracy. The optimal retraction distances for spheres and tubes that help the method achieve the second-order accuracy are found to be around 0.30 and -0.47 times of the lattice spacing, respectively. Simulations of the Stokes flow through a simple cubic lattice of rotational spheres indicate that the lift force produced by the Magnus effect can be very significant in view of the magnitude of the drag force when the practical rotating speed of the spheres is encountered. This finding

  5. A New Method of Embedded Fourth Order with Four Stages to Study Raster CNN Simulation

    R. Ponalagusamy; S. Senthilkumar


    A new Runge-Kutta (PK) fourth order with four stages embedded method with error control is presented in this paper for raster simulation in cellular neural network (CNN) environment. Through versatile algorithm, single layer/raster CNN array is implemented by incorporating the proposed technique. Simulation results have been obtained, and comparison has also been carried out to show the efficiency of the proposed numerical integration algorithm. The analytic expressions for local truncation error and global truncation error are derived. It is seen that the RK-embedded root mean square outperforms the RK-embedded Heronian mean and RK-embedded harmonic mean.

  6. Comparison of Optimal Homotopy Asymptotic and Adomian Decomposition Methods for a Thin Film Flow of a Third Grade Fluid on a Moving Belt

    Fazle Mabood


    Full Text Available We have investigated a thin film flow of a third grade fluid on a moving belt using a powerful and relatively new approximate analytical technique known as optimal homotopy asymptotic method (OHAM. The variation of velocity profile for different parameters is compared with the numerical values obtained by Runge-Kutta Fehlberg fourth-fifth order method and with Adomian Decomposition Method (ADM. An interesting result of the analysis is that the three terms OHAM solution is more accurate than five terms of the ADM solution and this thus confirms the feasibility of the proposed method.

  7. New Transcorrelated Method Improving the Feasibility of Explicitly Correlated Calculations

    Osamu Hino


    Full Text Available Abstract: We recently developed an explicitly correlated method using the transcorrelated Hamiltonian, which is preliminarily parameterized in such a way that the Coulomb repulsion is compensated at short inter-electronic distances. The extra part of the effective Hamiltonian features short-ranged, size-consistent, and state-universal. The localized and frozen nature of the correlation factor makes the enormous three-body interaction less important and enables us to bypass the complex nonlinear optimization. We review the basic strategy of the method mainly focusing on the applications to single-reference many electron theories using modified Møller-Plesset partitioning and biorthogonal orbitals. Benchmark calculations are performed for 10-electron systems with a series of basis sets.

  8. A Freestream-Preserving High-Order Finite-Volume Method for Mapped Grids with Adaptive-Mesh Refinement

    Guzik, S; McCorquodale, P; Colella, P


    A fourth-order accurate finite-volume method is presented for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space combined with detailed mechanisms for accommodating the adapting grids ensure that conservation is maintained and that the divergence of a constant vector field is always zero (freestream-preservation property). Advancement in time is achieved with a fourth-order Runge-Kutta method.

  9. Hybridizable discontinuous Galerkin projection methods for Navier-Stokes and Boussinesq equations

    Ueckermann, M. P.; Lermusiaux, P. F. J.


    Schemes for the incompressible Navier-Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit-Explicit Runge-Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed-element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.

  10. A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

    Liska, Sebastian; Colonius, Tim


    A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation.

  11. Vertical slice modelling of nonlinear Eady waves using a compatible finite element method

    Yamazaki, Hiroe; Shipton, Jemma; Cullen, Michael J. P.; Mitchell, Lawrence; Cotter, Colin J.


    A vertical slice model is developed for the Euler-Boussinesq equations with a constant temperature gradient in the direction normal to the slice (the Eady-Boussinesq model). The model is a solution of the full three-dimensional equations with no variation normal to the slice, which is an idealised problem used to study the formation and subsequent evolution of weather fronts. A compatible finite element method is used to discretise the governing equations. To extend the Charney-Phillips grid staggering in the compatible finite element framework, we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space. For the time discretisation, we solve the semi-implicit equations together with an explicit strong-stability-preserving Runge-Kutta scheme to all of the advection terms. The model reproduces several quasi-periodic lifecycles of fronts despite the presence of strong discontinuities. An asymptotic limit analysis based on the semi-geostrophic theory shows that the model solutions are converging to a solution in cross-front geostrophic balance. The results are consistent with the previous results using finite difference methods, indicating that the compatible finite element method is performing as well as finite difference methods for this test problem. We observe dissipation of kinetic energy of the cross-front velocity in the model due to the lack of resolution at the fronts, even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi-geostrophic limit solution.

  12. Explicitly Correlated Methods within the ccCA Methodology.

    Mahler, Andrew; Wilson, Angela K


    The prediction of energetic properties within "chemical accuracy" (1 kcal mol(-1) from well-established experiment) can be a major challenge in computational quantum chemistry due to the computational requirements (computer time, memory, and disk space) needed to achieve this level of accuracy. Methodologies such as coupled cluster with single, double, and perturbative triple excitations (CCSD(T)) combined with very large basis sets are often required to reach this level of accuracy. Unfortunately, such calculations quickly become cost prohibitive as system size increases. Our group has developed an ab initio composite method, the correlation consistent Composite Approach (ccCA), which enables such accuracy to be possible, on average, but at reduced computational cost as compared with CCSD(T) in combination with a large basis set. While ccCA has proven quite useful, computational bottlenecks still occur. In this study, the means to reduce the computational cost of ccCA without compromising accuracy by utilizing explicitly correlated methods within ccCA have been considered, and an alternative formulation is described.

  13. Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model

    Dehghan, Mehdi; Mohammadi, Vahid


    As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank-Nicolson scheme and another one is based on explicit Runge-Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.

  14. An improved algorithm of the fourth-order Runge-Kutta method and seismic wave-field simulation%四阶龙格-库塔方法的一种改进算法及地震波场模拟

    陈山; 杨顶辉; 邓小英



  15. Discontinuous Galerkin finite element method for the nonlinear hyperbolic problems with entropy-based artificial viscosity stabilization

    Zingan, Valentin Nikolaevich

    This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p > 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization.

  16. Advanced differential quadrature methods

    Zong, Zhi


    Modern Tools to Perform Numerical DifferentiationThe original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in applied mathematics, computational mechanics, and engineering have developed a range of innovative DQ-based methods to overcome these shortcomings. Advanced Differential Quadrature Methods explores new DQ methods and uses these methods to solve problems beyond the capabilities of the direct DQ method.After a basic introduction to the direct DQ method, the book presents a number of DQ methods, including complex DQ, triangular DQ, multi-scale DQ, variable order DQ, multi-domain DQ, and localized DQ. It also provides a mathematical compendium that summarizes Gauss elimination, the Runge-Kutta method, complex analysis, and more. The final chapter contains three codes written in the FORTRAN language, enabling readers to q...

  17. Models and numerical methods for time- and energy-dependent particle transport

    Olbrant, Edgar


    Particles passing through a medium can be described by the Boltzmann transport equation. Therein, all physical interactions of particles with matter are given by cross sections. We compare different analytical models of cross sections for photons, electrons and protons to state-of-the-art databases. The large dimensionality of the transport equation and its integro-differential form make it analytically difficult and computationally costly to solve. In this work, we focus on the following approximative models to the linear Boltzmann equation: (i) the time-dependent simplified P{sub N} (SP{sub N}) equations, (ii) the M{sub 1} model derived from entropy-based closures and (iii) a new perturbed M{sub 1} model derived from a perturbative entropy closure. In particular, an asymptotic analysis for SP{sub N} equations is presented and confirmed by numerical computations in 2D. Moreover, we design an explicit Runge-Kutta discontinuous Galerkin (RKDG) method to the M{sub 1} model of radiative transfer in slab geometry and construct a scheme ensuring the realizability of the moment variables. Among other things, M{sub 1} numerical results are compared with an analytical solution in a Riemann problem and the Marshak wave problem is considered. Additionally, we rigorously derive a new hierarchy of kinetic moment models in the context of grey photon transport in one spatial dimension. For the perturbed M{sub 1} model, we present numerical results known as the two beam instability or the analytical benchmark due to Su and Olson and compare them to the standard M{sub 1} as well as transport solutions.

  18. Efficient extrapolation methods for electro- and magnetoquasistatic field simulations

    M. Clemens


    Full Text Available In magneto- and electroquasi-static time domain simulations with implicit time stepping schemes the iterative solvers applied to the large sparse (non-linear systems of equations are observed to converge faster if more accurate start solutions are available. Different extrapolation techniques for such new time step solutions are compared in combination with the preconditioned conjugate gradient algorithm. Simple extrapolation schemes based on Taylor series expansion are used as well as schemes derived especially for multi-stage implicit Runge-Kutta time stepping methods. With several initial guesses available, a new subspace projection extrapolation technique is proven to produce an optimal initial value vector. Numerical tests show the resulting improvements in terms of computational efficiency for several test problems. In quasistatischen elektromagnetischen Zeitbereichsimulationen mit impliziten Zeitschrittverfahren zeigt sich, dass die iterativen Lösungsverfahren für die großen dünnbesetzten (nicht-linearen Gleichungssysteme schneller konvergieren, wenn genauere Startlösungen vorgegeben werden. Verschiedene Extrapolationstechniken werden für jeweils neue Zeitschrittlösungen in Verbindung mit dem präkonditionierten Konjugierte Gradientenverfahren vorgestellt. Einfache Extrapolationsverfahren basierend auf Taylorreihenentwicklungen werden ebenso benutzt wie speziell für mehrstufige implizite Runge-Kutta-Verfahren entwickelte Verfahren. Sind verschiedene Startlösungen verfügbar, so erlaubt ein neues Unterraum-Projektion- Extrapolationsverfahren die Konstruktion eines optimalen neuen Startvektors. Numerische Tests zeigen die aus diesen Verfahren resultierenden Verbesserungen der numerischen Effizienz.

  19. High Order A-stable Continuous General Linear Methods for Solution of Systems of Initial Value Problems in ODEs

    Dauda GuliburYAKUBU


    Full Text Available Accurate solutions to initial value systems of ordinary differential equations may be approximated efficiently by Runge-Kutta methods or linear multistep methods. Each of these has limitations of one sort or another. In this paper we consider, as a middle ground, the derivation of continuous general linear methods for solution of stiff systems of initial value problems in ordinary differential equations. These methods are designed to combine the advantages of both Runge-Kutta and linear multistep methods. Particularly, methods possessing the property of A-stability are identified as promising methods within this large class of general linear methods. We show that the continuous general linear methods are self-starting and have more ability to solve the stiff systems of ordinary differential equations, than the discrete ones. The initial value systems of ordinary differential equations are solved, for instance, without looking for any other method to start the integration process. This desirable feature of the proposed approach leads to obtaining very high accuracy of the solution of the given problem. Illustrative examples are given to demonstrate the novelty and reliability of the methods.

  20. Time integration algorithms for the two-dimensional Euler equations on unstructured meshes

    Slack, David C.; Whitaker, D. L.; Walters, Robert W.


    Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.

  1. Prediction of residual stress using explicit finite element method

    W.A. Siswanto


    Full Text Available This paper presents the residual stress behaviour under various values of friction coefficients and scratching displacement amplitudes. The investigation is based on numerical solution using explicit finite element method in quasi-static condition. Two different aeroengine materials, i.e. Super CMV (Cr-Mo-V and Titanium alloys (Ti-6Al-4V, are examined. The usage of FEM analysis in plate under normal contact is validated with Hertzian theoretical solution in terms of contact pressure distributions. The residual stress distributions along with normal and shear stresses on elastic and plastic regimes of the materials are studied for a simple cylinder-on-flat contact configuration model subjected to normal loading, scratching and followed by unloading. The investigated friction coefficients are 0.3, 0.6 and 0.9, while scratching displacement amplitudes are 0.05 mm, 0.10 mm and 0.20 mm respectively. It is found that friction coefficient of 0.6 results in higher residual stress for both materials. Meanwhile, the predicted residual stress is proportional to the scratching displacement amplitude, higher displacement amplitude, resulting in higher residual stress. It is found that less residual stress is predicted on Super CMV material compared to Ti-6Al-4V material because of its high yield stress and ultimate strength. Super CMV material with friction coefficient of 0.3 and scratching displacement amplitude of 0.10 mm is recommended to be used in contact engineering applications due to its minimum possibility of fatigue.

  2. Solution of the Michaelis-Menten equation using the decomposition method.

    Sonnad, Jagadeesh R; Goudar, Chetan T


    We present a low-order recursive solution to the Michaelis-Menten equation using the decomposition method. This solution is algebraic in nature and provides a simpler alternative to numerical approaches such as differential equation evaluation and root-solving techniques that are currently used to compute substrate concentration in the Michaelis-Menten equation. A detailed characterization of the errors in substrate concentrations computed from decomposition, Runge-Kutta, and bisection methods over a wide range of s(0) : K(m) values was made by comparing them with highly accurate solutions obtained using the Lambert W function. Our results indicated that solutions obtained from the decomposition method were usually more accurate than those from the corresponding classical Runge-Kutta methods. Moreover, these solutions required significantly fewer computations than the root-solving method. Specifically, when the stepsize was 0.1% of the total time interval, the computed substrate concentrations using the decomposition method were characterized by accuracies on the order of 10(-8) or better. The algebraic nature of the decomposition solution and its relatively high accuracy make this approach an attractive candidate for computing substrate concentration in the Michaelis-Menten equation.

  3. New transfer matrix method for long-period fiber gratings with coupled multiple cladding modes

    Guodong Wang; Yunjian Wang


    A new transfer matrix method for long-period fiber gratings with coupled multiple cladding modes is proposed and numerically characterized. The transmission spectra of uniform and non-uniform long-period fiber gratings are numerically characterized. The theoretical results excellently agree with the experimental measurements. Compared with commonly used methods, such as using the fourth-order adaptive step size control of the Runge-Kutta algorithm in solving the coupled mode equation, the new transfer matrix method exhibits a faster calculation speed.%@@ A new transfer matrix method for long-period fiber gratings with coupled multiple cladding modes is proposed and numerically characterized.The transmission spectra of uniform and non-uniform longperiod fiber gratings are numerically characterized.The theoretical results excellently agree with the experimental measurements.Compared with commonly used methods,such as using the fourth-order adaptive step size control of the Runge-Kutta algorithm in solving the coupled mode equation,the new transfer matrix method exhibits a faster calculation speed.

  4. Interlaminar stress analysis of dropped-ply laminated plates and shells by a mixed method. Ph.D. Thesis

    Harrison, Peter N.; Johnson, Eric R.; Starnes, James H., Jr.


    A mixed method of approximation based on Reissner's variational principle is developed for the linear analysis of interlaminar stresses in laminated composites, with special interest in laminates that contain terminated internal plies (dropped-ply laminates). Two models are derived, one for problems of generalized plane deformation and the other for the axisymmetric response of shells of revolution. A layerwise approach is taken in which the stress field is assumed with an explicit dependence on the thickness coordinate in each layer. The dependence of the stress field on the thickness coordinate is determined such that the three-dimensional equilibrium equations are satisfied by the approximation. The solution domain is reduced to one dimension by integration through the thickness. Continuity of tractions and displacements between layers is imposed. The governing two-point boundary value problem is composed of a system of both differential and algebraic equations (DAE's) and their associated boundary conditions. Careful evaluation of the system of DAE's was required to arrive at a form that allowed application of a one-step finite difference approximation. A two-stage Gauss implicit Runge-Kutta finite difference scheme was used for the solution because of its relatively high degree of accuracy. Patch tests of the two models revealed problems with solution accuracy for the axisymmetric model of a cylindrical shell loaded by internal pressure. Parametric studies of dropped-ply laminate characteristics and their influence on the interlaminar stresses were performed using the generalized plane deformation model. Eccentricity of the middle surface of the laminate through the ply drop-off was found to have a minimal effect on the interlaminar stresses under longitudinal compression, transverse tension, and in-plane shear. A second study found the stiffness change across the ply termination to have a much greater influence on the interlaminar stresses.

  5. An h-adaptive local discontinuous Galerkin method for the Navier-Stokes-Korteweg equations

    Tian, Lulu; Xu, Yan; Kuerten, J. G. M.; van der Vegt, J. J. W.


    In this article, we develop a mesh adaptation algorithm for a local discontinuous Galerkin (LDG) discretization of the (non)-isothermal Navier-Stokes-Korteweg (NSK) equations modeling liquid-vapor flows with phase change. This work is a continuation of our previous research, where we proposed LDG discretizations for the (non)-isothermal NSK equations with a time-implicit Runge-Kutta method. To save computing time and to capture the thin interfaces more accurately, we extend the LDG discretization with a mesh adaptation method. Given the current adapted mesh, a criterion for selecting candidate elements for refinement and coarsening is adopted based on the locally largest value of the density gradient. A strategy to refine and coarsen the candidate elements is then provided. We emphasize that the adaptive LDG discretization is relatively simple and does not require additional stabilization. The use of a locally refined mesh in combination with an implicit Runge-Kutta time method is, however, non-trivial, but results in an efficient time integration method for the NSK equations. Computations, including cases with solid wall boundaries, are provided to demonstrate the accuracy, efficiency and capabilities of the adaptive LDG discretizations.

  6. An investigation of GPU-based stiff chemical kinetics integration methods

    Curtis, Nicholas J; Sung, Chih-Jen


    A fifth-order implicit Runge-Kutta method and two fourth-order exponential integration methods equipped with Krylov subspace approximations were implemented for the GPU and paired with the analytical chemical kinetic Jacobian software pyJac. The performance of each algorithm was evaluated by integrating thermochemical state data sampled from stochastic partially stirred reactor simulations and compared with the commonly used CPU-based implicit integrator CVODE. We estimated that the implicit Runge-Kutta method running on a single GPU is equivalent to CVODE running on 12-38 CPU cores for integration of a single global integration time step of 1e-6 s with hydrogen and methane models. In the stiffest case studied---the methane model with a global integration time step of 1e-4 s---thread divergence and higher memory traffic significantly decreased GPU performance to the equivalent of CVODE running on approximately three CPU cores. The exponential integration algorithms performed more slowly than the implicit inte...

  7. An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

    Zhao Guo-Zhong; Yu Xi-Jun; Zhang Rong-Pei


    In this paper,Runge-Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical flux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.

  8. Generalización a Rn de algunos métodos de interpolación conocidos en ecuaciones diferenciales

    Vernor Arguedas Troyo


    Full Text Available We present the Runge-Kutta methods of several one-step levels in Rn , as well as algorithms in pseudo-code for the implicit and explicit methods. We study the problem of error control in Rn and we give numerical examples in tables or parameter schemata. Keywords: n-dimensional runge-Katta methods, explicit and implicit methods, interpolation, pseudo-algorithms, parametric schema.

  9. Optimal Runge-Kutta Schemes for High-order Spatial and Temporal Discretizations


    Discretizations 5a. CONTRACT NUMBER In-House 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Mundis , N., Edoh, A. and Sankaran, V. 5d...Schemes for High-order Spatial and Temporal Discretizations Nathan L. Mundis ∗ Ayaboe K. Edoh† Venkateswaran Sankaran‡ * ERC, Inc., †University of...the wave number being the parameter) are overlaid on the contour map of the amplification factor in the complex plane for the chosen temporal scheme

  10. Numerical methods for ordinary differential equations

    Butcher, John C


    In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author''s pioneering text is fully revised and updated to acknowledge many of these developments.  It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding.  Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler. Features of the book includeIntroductory work on differential and difference equations.A comprehensive introduction to the theory and practice of solving ordinary differential equations numeri...

  11. Calculation methods of the nuclear characteristics

    Dubovichenko, S B


    In the book the mathematical methods of nuclear cross sections and phases of elastic scattering, energy and characteristics of bound states in two- and three-particle nuclear systems, when the potentials of interaction contain not only central, but also tensor component, are presented. Are given the descriptions of the mathematical numerical calculation methods and computer programs in the algorithmic language "BASIC" for "Turbo Basic" of firm "Borland" for the computers of the type IBM PC AT. For the numerical solutions of the initial Schroedinger equations are used finite- difference and variational methods, and also method of Runge-Kutta with the automatic calling sequence on the assigned accuracy of results for the scattering phase shifts and binding energy. Is given the description not of the standard methods of solving the system of equations of Schroedinger to the bound states and the alternative to Schmidt's method, method of solution of the generalized matrix problem at the eigenvalues. The developed...

  12. Verifying Real-Time Systems using Explicit-time Description Methods

    Wang, Hao; 10.4204/EPTCS.13.6


    Timed model checking has been extensively researched in recent years. Many new formalisms with time extensions and tools based on them have been presented. On the other hand, Explicit-Time Description Methods aim to verify real-time systems with general untimed model checkers. Lamport presented an explicit-time description method using a clock-ticking process (Tick) to simulate the passage of time together with a group of global variables for time requirements. This paper proposes a new explicit-time description method with no reliance on global variables. Instead, it uses rendezvous synchronization steps between the Tick process and each system process to simulate time. This new method achieves better modularity and facilitates usage of more complex timing constraints. The two explicit-time description methods are implemented in DIVINE, a well-known distributed-memory model checker. Preliminary experiment results show that our new method, with better modularity, is comparable to Lamport's method with respect...

  13. Time Collocation Method for Structural Dynamic Problems

    TANG Chen; LUO Tao; YAN Haiqing; GU Xiaohui


    In order to achieve highly accurate and efficient numerical calculations of structural dynamics, time collocation method is presented. For a given time interval, the numerical solution of the method is approximated by a polynomial. The polynomial coefficients are evaluated by solving algebraic equation. Once the polynomial coefficients are evaluated, the numerical solutions at any time in the interval can be easily calculated. New formulae are derived for the polynomial coefficients,which are more practical and succinct than those previously given. Two structural dynamic equations are calculated by the proposed method. The numerical solutions are compared with the traditional fourth-order Runge-Kutta method. The results show that the method proposed is highly accurate and computationally efficient. In addition, an important advantage of the method is the simplicity in software programming.

  14. Numerical methods of microirrigation lateral design

    Kettab A.


    Full Text Available The present work contributes to the hydraulic analysis of the lateral microirrigation by using the numerical methods: the control volumes method “CVM” and the Runge-Kutta method “RK4”. These methods are relatively simple to manipulate and agree to the use of the partial differential equations of the first order. The CVM method warrants to follow the hydraulic phenomenon step by step and facilitates iterative development; whereas, the RK4 method is used in the integration and the solution of the differential equations system. The risk of divergence, as the slowness of the computation is avoided by the recourse to the interpolation using the polynomial of Lagrange in order to accelerate the convergence toward the solution. The models of calculation used have the advantage to be simple, fast, precise, and allow their extension to large microirrigation network.

  15. Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics

    Wu, Shen R


    A systematic introduction to the theories and formulations of the explicit finite element method As numerical technology continues to grow and evolve with industrial applications, understanding the explicit finite element method has become increasingly important, particularly in the areas of crashworthiness, metal forming, and impact engineering. Introduction to the Explicit FiniteElement Method for Nonlinear Transient Dynamics is the first book to address specifically what is now accepted as the most successful numerical tool for nonlinear transient dynamics. The book aids readers in master

  16. A CAA Primer for Practicing Engineers


    is slightly dispersed . The cause is temporal discretization. For comparison purpose, Fig. 3.5c shows the computed waveform using the LDD Runge-Kutta...amplification factor and the amplification factor when Runge-Kutta scheme was used. They called their scheme low- dissipation low- dispersion ( LDD ) Runge...2.5.1 Single Time Step Method : Runge-Kutta Scheme 2.5.2 Four-Level DRP Scheme 3. NUMERICAL DISPERSION AND DISSIPATION OF HIGH-ORDER

  17. Explicit finite difference methods for the delay pseudoparabolic equations.

    Amirali, I; Amiraliyev, G M; Cakir, M; Cimen, E


    Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

  18. Explicit local time-stepping methods for time-dependent wave propagation

    Grote, Marcus


    Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leap-frog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching scheme...

  19. Numerical methods in vehicle system dynamics: state of the art and current developments

    Arnold, M.; Burgermeister, B.; Führer, C.; Hippmann, G.; Rill, G.


    Robust and efficient numerical methods are an essential prerequisite for the computer-based dynamical analysis of engineering systems. In vehicle system dynamics, the methods and software tools from multibody system dynamics provide the integration platform for the analysis, simulation and optimisation of the complex dynamical behaviour of vehicles and vehicle components and their interaction with hydraulic components, electronical devices and control structures. Based on the principles of classical mechanics, the modelling of vehicles and their components results in nonlinear systems of ordinary differential equations (ODEs) or differential-algebraic equations (DAEs) of moderate dimension that describe the dynamical behaviour in the frequency range required and with a level of detail being characteristic of vehicle system dynamics. Most practical problems in this field may be transformed to generic problems of numerical mathematics like systems of nonlinear equations in the (quasi-)static analysis and explicit ODEs or DAEs with a typical semi-explicit structure in the dynamical analysis. This transformation to mathematical standard problems allows to use sophisticated, freely available numerical software that is based on well approved numerical methods like the Newton-Raphson iteration for nonlinear equations or Runge-Kutta and linear multistep methods for ODE/DAE time integration. Substantial speed-ups of these numerical standard methods may be achieved exploiting some specific structure of the mathematical models in vehicle system dynamics. In the present paper, we follow this framework and start with some modelling aspects being relevant from the numerical viewpoint. The focus of the paper is on numerical methods for static and dynamic problems, including software issues and a discussion which method fits best for which class of problems. Adaptive components in state-of-the-art numerical software like stepsize and order control in time integration are

  20. Explicit and Implicit Kinetic Streamlined-Upwind Petrov Galerkin Method for Hyperbolic Partial Differential Equations

    Jagtap, Ameya Dilip


    A novel explicit and implicit Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) scheme is presented for hyperbolic equations such as Burgers equation and compressible Euler equations. The proposed scheme performs better than the original SUPG stabilized method in multi-dimensions. To demonstrate the numerical accuracy of the scheme, various numerical experiments have been carried out for 1D and 2D Burgers equation as well as for 1D and 2D Euler equations using Q4 and T3 elements. Furthermore, spectral stability analysis is done for the explicit 2D formulation. Finally, a comparison is made between explicit and implicit versions of the KSUPG scheme.


    MEI Shu-li; LU Qi-shao; ZHANG Sen-wen; JIN Li


    The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations (ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.

  2. Three-dimensional beam propagation method based on the variable transformed Galerkin's method

    XIAO Jinbiao; SUN Xiaohan; ZHANG Mingde


    A novel three-dimensional beam propagation method (BPM) based on the variable transformed Galerkin's method is introduced for simulating optical field propagation in three-dimensional dielectric structures. The infinite Cartesian x-y plane is mapped into a unit square by a tangent-type function transformation. Consequently, the infinite region problem is converted into the finite region problem. Thus, the boundary truncation is eliminated and the calculation accuracy is promoted. The three-dimensional BPM basic equation is reduced to a set of first-order ordinary differential equations through sinusoidal basis function, which fits arbitrary cladding optical waveguide, then direct solution of the resulting equations by means of the Runge-Kutta method. In addition,the calculation is efficient due to the small matrix derived from the present technique.Both z-invariant and z-variant examples are considered to test both the accuracy and utility of this approach.

  3. An Efficient Explicit-time Description Method for Timed Model Checking

    Wang, Hao; 10.4204/EPTCS.14.6


    Timed model checking, the method to formally verify real-time systems, is attracting increasing attention from both the model checking community and the real-time community. Explicit-time description methods verify real-time systems using general model constructs found in standard un-timed model checkers. Lamport proposed an explicit-time description method using a clock-ticking process (Tick) to simulate the passage of time together with a group of global variables to model time requirements. Two methods, the Sync-based Explicit-time Description Method using rendezvous synchronization steps and the Semaphore-based Explicit-time Description Method using only one global variable were proposed; they both achieve better modularity than Lamport's method in modeling the real-time systems. In contrast to timed automata based model checkers like UPPAAL, explicit-time description methods can access and store the current time instant for future calculations necessary for many real-time systems, especially those with p...

  4. Stiff modes in spinvalve simulations with OOMMF

    Mitropoulos, Spyridon [Department of Computer and Informatics Engineering, TEI of Eastern Macedonia and Thrace, Kavala (Greece); Tsiantos, Vassilis, E-mail: [Department of Electrical Engineering, TEI of Eastern Macedonia and Thrace, Kavala, 65404 Greece (Greece); Ovaliadis, Kyriakos [Department of Electrical Engineering, TEI of Eastern Macedonia and Thrace, Kavala, 65404 Greece (Greece); Kechrakos, Dimitris [Department of Education, ASPETE, Heraklion, Athens (Greece); Donahue, Michael [Applied and Computational Mathematics Division, NIST, Gaithersburg, MD (United States)


    Micromagnetic simulations are an important tool for the investigation of magnetic materials. Micromagnetic software uses various techniques to solve differential equations, partial or ordinary, involved in the dynamic simulations. Euler, Runge-Kutta, Adams, and BDF (Backward Differentiation Formulae) are some of the methods used for this purpose. In this paper, spinvalve simulations are investigated. Evidence is presented showing that these systems have stiff modes, and that implicit methods such as BDF are more effective than explicit methods in such cases.

  5. A new approach to constructing efficient stiffly accurate EPIRK methods

    Rainwater, G.; Tokman, M.


    The structural flexibility of the exponential propagation iterative methods of Runge-Kutta type (EPIRK) enables construction of particularly efficient exponential time integrators. While the EPIRK methods have been shown to perform well on stiff problems, all of the schemes proposed up to now have been derived using classical order conditions. In this paper we extend the stiff order conditions and the convergence theory developed for the exponential Rosenbrock methods to the EPIRK integrators. We derive stiff order conditions for the EPIRK methods and develop algorithms to solve them to obtain specific schemes. Moreover, we propose a new approach to constructing particularly efficient EPIRK integrators that are optimized to work with an adaptive Krylov algorithm. We use a set of numerical examples to illustrate the computational advantages that the newly constructed EPIRK methods offer compared to previously proposed exponential integrators.

  6. Numerical Methods and Comparisons for 1D and Quasi 2D Streamer Propagation Models

    Huang, Mengmin; Guan, Huizhe; Zeng, Rong


    In this work, we propose four different strategies to simulate the one-dimensional (1D) and quasi two-dimensional (2D) model for streamer propagation. Each strategy involves of one numerical method for solving Poisson's equation and another method for solving continuity equations in the models, and a total variation diminishing three-stage Runge-Kutta method in temporal discretization. The numerical methods for Poisson's equation include finite volume method, discontinuous Galerkin methods, mixed finite element method and least-squared finite element method. The numerical method for continuity equations is chosen from the family of discontinuous Galerkin methods. The accuracy tests and comparisons show that all of these four strategies are suitable and competitive in streamer simulations from the aspects of accuracy and efficiency. By applying any strategy in real simulations, we can study the dynamics of streamer propagations and influences due to the change of parameters in both of 1D and quasi 2D models. T...

  7. Semi-implicit spectral deferred correction methods for ordinary differential equations

    Minion, Michael L.


    A semi-implicit formulation of the method of spectral deferred corrections (SISDC) for ordinary differential equations with both stiff and non-stiff terms is presented. Several modifications and variations to the original spectral deferred corrections method by Dutt, Greengard, and Rokhlin concerning the choice of integration points and the form of the correction iteration are presented. The stability and accuracy of the resulting ODE methods are explored analytically and numerically. The SISDC methods are intended to be combined with the method of lines approach to yield a flexible framework for creating higher-order semi-implicit methods for partial differential equations. A discussion and numerical examples of the SISDC method applied to advection-diffusion type equations are included. The results suggest that higher-order SISDC methods are more efficient than semi-implicit Runge-Kutta methods for moderately stiff problems in terms of accuracy per function evaluation.

  8. The Piecewise Cubic Method (PCM) for computational fluid dynamics

    Lee, Dongwook; Faller, Hugues; Reyes, Adam


    We present a new high-order finite volume reconstruction method for hyperbolic conservation laws. The method is based on a piecewise cubic polynomial which provides its solutions a fifth-order accuracy in space. The spatially reconstructed solutions are evolved in time with a fourth-order accuracy by tracing the characteristics of the cubic polynomials. As a result, our temporal update scheme provides a significantly simpler and computationally more efficient approach in achieving fourth order accuracy in time, relative to the comparable fourth-order Runge-Kutta method. We demonstrate that the solutions of PCM converges at fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme on a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.

  9. The Piecewise Cubic Method (PCM) for Computational Fluid Dynamics

    Lee, Dongwook; Reyes, Adam


    We present a new high-order finite volume reconstruction method for hyperbolic conservation laws. The method is based on a piecewise cubic polynomial which provides its solutions a fifth-order accuracy in space. The spatially reconstructed solutions are evolved in time with a fourth-order accuracy by tracing the characteristics of the cubic polynomials. As a result, our temporal update scheme provides a significantly simpler and computationally more efficient approach in achieving fourth order accuracy in time, relative to the comparable fourth-order Runge-Kutta method. We demonstrate that the solutions of PCM converges in fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme in a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.

  10. Improved stochastic approximation methods for discretized parabolic partial differential equations

    Guiaş, Flavius


    We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).


    李庆宏; 罗亮生; 吴新元


    This paper presents a two-step explicit method of order four for solving aclass of linear periodic initial value problems. At each computational step, only tworight function evaluations and one derivative evaluation are employed. Basing on aspecial vector operation, the method can be extended to the vector-applicable in multi-dimensional space.

  12. Transient Analysis of Hysteresis Queueing Model Using Matrix Geometric Method

    Wajiha Shah


    Full Text Available Various analytical methods have been proposed for the transient analysis of a queueing system in the scalar domain. In this paper, a vector domain based transient analysis is proposed for the hysteresis queueing system with internal thresholds for the efficient and numerically stable analysis. In this system arrival rate of customer is controlled through the internal thresholds and the system is analyzed as a quasi-birth and death process through matrix geometric method with the combination of vector form Runge-Kutta numerical procedure which utilizes the special matrices. An arrival and service process of the system follows a Markovian distribution. We analyze the mean number of customers in the system when the system is in transient state against varying time for a Markovian distribution. The results show that the effect of oscillation/hysteresis depends on the difference between the two internal threshold values.

  13. Introduction to numerical and analytical methods with Matlab for engineers and scientists

    Bober, William


    The text covers useful numerical methods, including interpolation, Simpson’s rule on integration, the Gauss elimination method for solving systems of linear algebraic equations, the Runge-Kutta method for solving ordinary differential equations, and the search method in combination with the bisection method for obtaining the roots of transcendental and polynomial equations. It also highlights MATLAB’s built-in functions. These include interp1 function, the quad and dblquad functions, the inv function, the ode45 function, the fzero function, and many others. The second half of the text covers more advanced topics, including the iteration method for solving pipe flow problems, the Hardy-Cross method for solving flow rates in a pipe network, separation of variables for solving partial differential equations, and the use of Laplace transforms to solve both ordinary and partial differential equations.

  14. Explicit One-Step P-Stable Methods for Second Order Periodic Initial Value Problems

    Qinghong Li; Yongzhong Song


    In this paper, we present an explicit one-step method for solving periodic initial value problems of second order ordinary differential equations. The method is P-stable, and of first algebraic order and high phase-lag order. To improve the algebraic order, we give a composition second order scheme with the proposed method and its adjoint. We report some numerical results to illustrate the efficiency of our methods.

  15. Multisymplectic implicit and explicit methods for Klein-Gordon-Schr(o)dinger equations

    Cai Jia-Xiang; Yang Bin; Liang Hua


    We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schr(o)dinger equations.We prove that the implicit method satisfies the charge conservation law exactly.Both methods provide accurate solutions in long-time computations and simulate the soliton collision well.The numerical results show the abilities of the two methods in preserving the charge,energy,and momentum conservation laws.

  16. High-order implicit residual smoothing time scheme for direct and large eddy simulations of compressible flows

    Cinnella, P.; Content, C.


    Restrictions on the maximum allowable time step of explicit time integration methods for direct and large eddy simulations of compressible turbulent flows at high Reynolds numbers can be very severe, because of the extremely small space steps used close to solid walls to capture tiny and elongated boundary layer structures. A way of increasing stability limits is to use implicit time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. In quest for an implicit time scheme for scale-resolving simulations providing the best possible compromise between these opposite requirements, we develop a Runge-Kutta implicit residual smoothing (IRS) scheme of fourth-order accuracy, based on a bilaplacian operator. The implicit operator involves the inversion of scalar pentadiagonal systems, for which efficient parallel algorithms are available. The proposed method is assessed against two explicit and two implicit time integration techniques in terms of computational cost required to achieve a threshold level of accuracy. Precisely, the proposed time scheme is compared to four-stages and six-stages low-storage Runge-Kutta method, to the second-order IRS and to a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations. Numerical results show that the proposed IRS scheme leads to reductions in computational time by a factor 3 to 5 for an accuracy comparable to that of the corresponding explicit Runge-Kutta scheme.

  17. An explicit Lagrangian finite element method for free-surface weakly compressible flows

    Cremonesi, Massimiliano; Meduri, Simone; Perego, Umberto; Frangi, Attilio


    In the present work, an explicit finite element approach to the solution of the Lagrangian formulation of the Navier-Stokes equations for weakly compressible fluids or fluid-like materials is investigated. The introduction of a small amount of compressibility is shown to allow for the formulation of a fast and robust explicit solver based on a particle finite element method. Newtonian and Non-Newtonian Bingham laws are considered. A barotropic equation of state completes the model relating pressure and density fields. The approach has been validated through comparison with experimental tests and numerical simulations of free surface fluid problems involving water and water-soil mixtures.

  18. Performance of Several High Order Numerical Methods for Supersonic Combustion

    Sjoegreen, Bjoern; Yee, H. C.; Don, Wai Sun; Mansour, Nagi N. (Technical Monitor)


    The performance of two recently developed numerical methods by Yee et al. and Sjoegreen and Yee using postprocessing nonlinear filters is examined for a 2-D multiscale viscous supersonic react-live flow. These nonlinear filters can improve nonlinear instabilities and at the same time can capture shock/shear waves accurately. They do not, belong to the class of TVD, ENO or WENO schemes. Nevertheless, they combine stable behavior at discontinuities and detonation without smearing the smooth parts of the flow field. For the present study, we employ a fourth-order Runge-Kutta in time and a sixth-order non-dissipative spatial base scheme for the convection and viscous terms. We denote the resulting nonlinear filter schemes ACM466-RK4 and WAV66-RK4.

  19. Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations

    Bonito, Andrea


    We establish the L2-stability of an entropy viscosity technique applied to nonlinear scalar conservation equations. First-and second-order explicit time-stepping techniques using continuous finite elements in space are considered. The method is shown to be stable independently of the polynomial degree of the space approximation under the standard CFL condition. © 2013 American Mathematical Society.

  20. A numerical study of 2D detonation waves with adaptive finite volume methods on unstructured grids

    Hu, Guanghui


    In this paper, a framework of adaptive finite volume solutions for the reactive Euler equations on unstructured grids is proposed. The main ingredients of the algorithm include a second order total variation diminishing Runge-Kutta method for temporal discretization, and the finite volume method with piecewise linear solution reconstruction of the conservative variables for the spatial discretization in which the least square method is employed for the reconstruction, and weighted essentially nonoscillatory strategy is used to restrain the potential numerical oscillation. To resolve the high demanding on the computational resources due to the stiffness of the system caused by the reaction term and the shock structure in the solutions, the h-adaptive method is introduced. OpenMP parallelization of the algorithm is also adopted to further improve the efficiency of the implementation. Several one and two dimensional benchmark tests on the ZND model are studied in detail, and numerical results successfully show the effectiveness of the proposed method.

  1. Extended Parker-Sochacki method for Michaelis-Menten enzymatic reaction model.

    Abdelrazik, Ismail M; Elkaranshawy, Hesham A


    In this article, a new approach--namely, the extended Parker-Sochacki method (EPSM)--is presented for solving the Michaelis-Menten nonlinear enzymatic reaction model. The Parker-Sochacki method (PSM) is combined with a new resummation method called the Sumudu-Padé resummation method to obtain approximate analytical solutions for the model. The obtained solutions by the proposed approach are compared with the solutions of PSM and the Runge-Kutta numerical method (RKM). The comparison proves the practicality, efficiency, and correctness of the presented approach. It serves as a basis for solving other nonlinear biochemical reaction models in the future. Copyright © 2015 Elsevier Inc. All rights reserved.

  2. Vibration analysis of nonlinear systems with the bilinear hysteretic oscillator by using incremental harmonic balance method

    Xiong, Huai; Kong, Xianren; Li, Haiqin; Yang, Zhenguo


    This paper considers dynamics of bilinear hysteretic systems, which are widely used for vibration control and vibration absorption such as magneto-rheological damper, metal-rubber. The method of incremental harmonic balance (IHB) technique that hysteresis is considered in the corrective term is improved in order to determine periodic solutions of bilinear hysteretic systems. The improved continuation method called two points tracing algorithm which is stable to the turning point makes the calculation more efficient for tracing amplitude-frequency response. Precise Hsu's method for analysing the stability of periodic solutions is introduced. The effects of different parameters of bilinear hysteretic oscillator on the response are discussed numerically. Some numerical simulations of considered bilinear hysteretic systems, including a single DOF and a 2DOF system, are effectively obtained by the modified IHB method and the results compare very well with the 4-oder Runge-Kutta method.

  3. Numerical methods for ordinary differential equations in the 20th century

    Butcher, J. C.


    Numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons. The first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of Bashforth and Adams for linear multistep methods and Runge for Runge-Kutta methods. Other reasons, which of course apply to numerical analysis in general, are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics. This survey paper follows many of the main strands in the developments of these methods, both for general problems, stiff systems, and for many of the special problem types that have been gaining in significance as the century draws to an end.

  4. A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

    Wu, Kailiang; Tang, Huazhong; Xiu, Dongbin


    This paper is concerned with generalized polynomial chaos (gPC) approximation for first-order quasilinear hyperbolic systems with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Then the gPC stochastic Galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is discretized by a path-conservative finite volume WENO scheme in space and a third-order total variation diminishing Runge-Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional gPC Galerkin system is carried over via an operator splitting technique. Several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

  5. 3D level set methods for evolving fronts on tetrahedral meshes with adaptive mesh refinement

    Morgan, Nathaniel R.; Waltz, Jacob I.


    The level set method is commonly used to model dynamically evolving fronts and interfaces. In this work, we present new methods for evolving fronts with a specified velocity field or in the surface normal direction on 3D unstructured tetrahedral meshes with adaptive mesh refinement (AMR). The level set field is located at the nodes of the tetrahedral cells and is evolved using new upwind discretizations of Hamilton-Jacobi equations combined with a Runge-Kutta method for temporal integration. The level set field is periodically reinitialized to a signed distance function using an iterative approach with a new upwind gradient. The details of these level set and reinitialization methods are discussed. Results from a range of numerical test problems are presented.

  6. Novel simulation method for fiber Bragg grating under inhomogeneous strain fields

    YUN Bin-feng; LU Chang-gui; WANG Zhu-yuan; WANG Yi-ping; CUI Yi-ping


    The spectra of fiber Bragg grating (FBG) in inhomogeneous strain fields are distorted due to its inhomogeneity of both the periods and the effective refractive index. The couple mode theory and the Runge-Kutta method can be employed for exact simulation of the spectrum of Bragg grating in such field, but the convergence speed is slow. On the other hand, although the transfer matrix method could be used with higher convergence speed, the precision is poor because of the neglect of the grads of strain change. By improving the FBG equivalent period, a novel simulation method based on a modified transfer matrix method is proposed, which has the advantage of quick-convergence as well as good accuracy.

  7. Comparison of explicitly correlated local coupled-cluster methods with various choices of virtual orbitals.

    Krause, Christine; Werner, Hans-Joachim


    Explicitly correlated local coupled-cluster (LCCSD-F12) methods with pair natural orbitals (PNOs), orbital specific virtual orbitals (OSVs), and projected atomic orbitals (PAOs) are compared. In all cases pair-specific virtual subspaces (domains) are used, and the convergence of the correlation energy as a function of the domain sizes is studied. Furthermore, the performance of the methods for reaction energies of 52 reactions involving 58 small and medium sized molecules is investigated. It is demonstrated that for all choices of virtual orbitals much smaller domains are needed in the explicitly correlated methods than without the explicitly correlated terms, since the latter correct a large part of the domain error, as found previously. For PNO-LCCSD-F12 with VTZ-F12 basis sets on the average only 20 PNOs per pair are needed to obtain reaction energies with a root mean square deviation of less than 1 kJ mol(-1) from complete basis set estimates. With OSVs or PAOs at least 4 times larger domains are needed for the same accuracy. A new hybrid method that combines the advantages of the OSV and PNO methods is proposed and tested. While in the current work the different local methods are only simulated using a conventional CCSD program, the implications for low-order scaling local implementations of the various methods are discussed.

  8. Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System

    M. S. H. Chowdhury


    Full Text Available Finding accurate solution of chaotic system by using efficient existing numerical methods is very hard for its complex dynamical behaviors. In this paper, the multistage homotopy-perturbation method (MHPM is applied to the Chaotic Genesio system. The MHPM is a simple reliable modification based on an adaptation of the standard homotopy-perturbation method (HPM. The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chaotic Genesio system. Numerical comparisons between the MHPM and the classical fourth-order Runge-Kutta (RK4 solutions are made. The results reveal that the new technique is a promising tool for the nonlinear chaotic systems of ordinary differential equations.



    A high-resolution finite volume numerical method for solving the shallow water equations is developed in this paper. In order to extend finite difference TVD scheme to finite volume method, a new geometry and topology of control bodies is defined by considering the corresponding relationships between nodes and elements. This solver is implemented on arbitrary quadrilateral meshes and their satellite elements, and based on a second-order hybrid type of TVD scheme in space discretization and a two-step Runge-Kutta method in time discretization. Then it is used to deal with two typical dam-break problems and very satisfactory results are obtained comparied with other numerical solutions. It can be considered as an efficient implement for the computation of shallow water problems, especially concerning those having discontinuities, subcritical and supercritical flows and complex geometries.

  10. New version of Optimal Homotopy Asymptotic Method for the solution of nonlinear boundary value problems in finite and infinite intervals

    Liaqat Ali


    Full Text Available In this research work a new version of Optimal Homotopy Asymptotic Method is applied to solve nonlinear boundary value problems (BVPs in finite and infinite intervals. It comprises of initial guess, auxiliary functions (containing unknown convergence controlling parameters and a homotopy. The said method is applied to solve nonlinear Riccati equations and nonlinear BVP of order two for thin film flow of a third grade fluid on a moving belt. It is also used to solve nonlinear BVP of order three achieved by Mostafa et al. for Hydro-magnetic boundary layer and micro-polar fluid flow over a stretching surface embedded in a non-Darcian porous medium with radiation. The obtained results are compared with the existing results of Runge-Kutta (RK-4 and Optimal Homotopy Asymptotic Method (OHAM-1. The outcomes achieved by this method are in excellent concurrence with the exact solution and hence it is proved that this method is easy and effective.

  11. Single Alternating Group Explicit (SAGE) Method for Electrochemical Finite Difference Digital Simulation

    DENG,Zhao-Xiang(邓兆祥); LIN,Xiang-Qin(林祥钦); TONG,Zhong-Hua(童中华)


    The four different schemes of Group Explicit Method (GEM): GER, GEL, SAGE and DAGE have been claimed to be unstable when employed for electrochemical digital simulations with large model diffusion coefficient DM@ However, in this investigation, in spite of the conditional stability of GER and GEL, the SAGE scheme, which is a combination of GEL and GER, was found to be unconditionally stable when used for simulations of electrochemical reaction-diffusions and had a performance comparable with or even better than the Fast Quasi Explicit Finite Difference Method (FQEFD) in srme aspects. Corresponding differential equations of SAGE scheme for digital simulations of various electrochemical mechanisms with both uniform and exponentially expanded space units were established. The effectiveness of the SAGE method was further demonstrated by the simulations of an EC and a catalytic mechanism with very large homogoneous rate constants.

  12. Post-Flight Trajectory Reconstruction of a Maneuvering Reentry Vehicle from Radar Measurements


    8217. *’.÷ ...... . Runge-Kutta method modified by Nystroih to handle the second " -"order equation. Kreyszig (Ref 21) calls this the Runge- Kutta...Prediction Theory," J. Basic Eng Vol 83, 1961, pp 95-108. 21. Kreyszig , E. Advanced Engineering Mathematics (FourthI ~. Edition), John Wiley and Sons

  13. The time-saving numerical method for GPS/MET observation operator

    李树勇; 王斌; 邹晓蕾; 刘辉


    The global positioning system (GPS) ray-shooting method is a self-sufficient observation operator inGPS/MET (meteorology) data variational assimilation linking up the GPS observation data and the atmosphere state vari-ables. But it cannot be applied to data assimilation and operational prediction so far because of huge computations. In order to reduce the amount of computation, a 2-order time-saving symplectic scheme is used to solve the equations of the GPS ray trajectory, due to its separable Hamiltonian nature, and good results are achieved. Not only does it save 75 % of CPU time taken by the old GPS ray-shooting model with 4th-order Runge-Kutta method , but also it improves the sim-ulation accuracy to some extent.

  14. The construction of arbitrary order ERKN methods based on group theory for solving oscillatory Hamiltonian systems with applications

    Mei, Lijie; Wu, Xinyuan


    In general, extended Runge-Kutta-Nyström (ERKN) methods are more effective than traditional Runge-Kutta-Nyström (RKN) methods in dealing with oscillatory Hamiltonian systems. However, the theoretical analysis for ERKN methods, such as the order conditions, the symplectic conditions and the symmetric conditions, becomes much more complicated than that for RKN methods. Therefore, it is a bottleneck to construct high-order ERKN methods efficiently. In this paper, we first establish the ERKN group Ω for ERKN methods and the RKN group G for RKN methods, respectively. We then rigorously show that ERKN methods are a natural extension of RKN methods, that is, there exists an epimorphism η of the ERKN group Ω onto the RKN group G. This epimorphism gives a global insight into the structure of the ERKN group by the analysis of its kernel and the corresponding RKN group G. Meanwhile, we establish a particular mapping φ of G into Ω so that each image element is an ideal representative element of the congruence class in Ω. Furthermore, an elementary theoretical analysis shows that this map φ can preserve many structure-preserving properties, such as the order, the symmetry and the symplecticity. From the epimorphism η together with its section φ, we may gain knowledge about the structure of the ERKN group Ω via the RKN group G. In light of the theoretical analysis of this paper, we obtain high-order structure-preserving ERKN methods in an effective way for solving oscillatory Hamiltonian systems. Numerical experiments are carried out and the results are very promising, which strongly support our theoretical analysis presented in this paper.

  15. Approximation of acoustic waves by explicit Newmark's schemes and spectral element methods

    Zampieri, Elena; Pavarino, Luca F.


    A numerical approximation of the acoustic wave equation is presented. The spatial discretization is based on conforming spectral elements, whereas we use finite difference Newmark's explicit integration schemes for the temporal discretization. A rigorous stability analysis is developed for the discretized problem providing an upper bound for the time step [Delta]t. We present several numerical results concerning stability and convergence properties of the proposed numerical methods.


    Carsten Carstensen; Joscha Gedicke; Donsub Rim


    The elementary analysis of this paper presents explicit expressions of the constants in the a priori error estimates for the lowest-order Courant,Crouzeix-Raviart nonconforming and Raviart-Thomas mixed finite element methods in the Poisson model problem.The three constants and their dependences on some maximal angle in the triangulation are indeed all comparable and allow accurate a priori error control.


    Jin-chao Xu; Lung-an Ying


    An explicit upwind finite element method is given for the numerical computation to multi-dimensional scalar conservation laws. It is proved that this scheme is consistent to the equation and monotone, and the approximate solution satisfies discrete entropy inequality.To guarantee the limit of approximate solutions to be a measure valued solution, we prove an energy estimate. Then the Lp strong convergence of this scheme is proved.


    Hua-zhong Tang; Hua-mu Wu


    This paper continues to construct and study the explicit compact (EC) schemes for conservation laws. First, we extend STCE/SE method on non-staggered grid, which has same well resolution as one in [1], and just requires half of the com- putational works. Then, we consider some constructions of the EC schemes for two-dimensional conservation laws, and some 1D and 2D numerical experiments are also given.

  19. Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

    Gurhan Gurarslan


    Full Text Available This study aims to produce numerical solutions of one-dimensional advection-diffusion equation using a sixth-order compact difference scheme in space and a fourth-order Runge-Kutta scheme in time. The suggested scheme here has been seen to be very accurate and a relatively flexible solution approach in solving the contaminant transport equation for Pe≤5. For the solution of the present equation, the combined technique has been used instead of conventional solution techniques. The accuracy and validity of the numerical model are verified through the presented results and the literature. The computed results showed that the use of the current method in the simulation is very applicable for the solution of the advection-diffusion equation. The present technique is seen to be a very reliable alternative to existing techniques for these kinds of applications.



    The resolution of differential games often concerns the difficult problem of two points border value (TPBV), then ascribe linear quadratic differential game to Hamilton system. To Hamilton system, the algorithm of symplectic geometry has the merits of being able to copy the dynamic structure of Hamilton system and keep the measure of phase plane. From the viewpoint of Hamilton system, the symplectic characters of linear quadratic differential game were probed; as a try, Symplectic-Runge-Kutta algorithm was presented for the resolution of infinite horizon linear quadratic differential game. An example of numerical calculation was given, and the result can illuminate the feasibility of this method. At the same time, it embodies the fine conservation characteristics of symplectic algorithm to system energy.

  1. A Parallel Implicit Reconstructed Discontinuous Galerkin Method for Compressible Flows on Hybrid Grids

    Xia, Yidong

    The objective this work is to develop a parallel, implicit reconstructed discontinuous Galerkin (RDG) method using Taylor basis for the solution of the compressible Navier-Stokes equations on 3D hybrid grids. This third-order accurate RDG method is based on a hierarchical weighed essentially non- oscillatory reconstruction scheme, termed as HWENO(P1P 2) to indicate that a quadratic polynomial solution is obtained from the underlying linear polynomial DG solution via a hierarchical WENO reconstruction. The HWENO(P1P2) is designed not only to enhance the accuracy of the underlying DG(P1) method but also to ensure non-linear stability of the RDG method. In this reconstruction scheme, a quadratic polynomial (P2) solution is first reconstructed using a least-squares approach from the underlying linear (P1) discontinuous Galerkin solution. The final quadratic solution is then obtained using a Hermite WENO reconstruction, which is necessary to ensure the linear stability of the RDG method on 3D unstructured grids. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the non-linear stability of the RDG method. The parallelization in the RDG method is based on a message passing interface (MPI) programming paradigm, where the METIS library is used for the partitioning of a mesh into subdomain meshes of approximately the same size. Both multi-stage explicit Runge-Kutta and simple implicit backward Euler methods are implemented for time advancement in the RDG method. In the implicit method, three approaches: analytical differentiation, divided differencing (DD), and automatic differentiation (AD) are developed and implemented to obtain the resulting flux Jacobian matrices. The automatic differentiation is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as

  2. Experiments with explicit filtering for LES using a finite-difference method

    Lund, T. S.; Kaltenbach, H. J.


    The equations for large-eddy simulation (LES) are derived formally by applying a spatial filter to the Navier-Stokes equations. The filter width as well as the details of the filter shape are free parameters in LES, and these can be used both to control the effective resolution of the simulation and to establish the relative importance of different portions of the resolved spectrum. An analogous, but less well justified, approach to filtering is more or less universally used in conjunction with LES using finite-difference methods. In this approach, the finite support provided by the computational mesh as well as the wavenumber-dependent truncation errors associated with the finite-difference operators are assumed to define the filter operation. This approach has the advantage that it is also 'automatic' in the sense that no explicit filtering: operations need to be performed. While it is certainly convenient to avoid the explicit filtering operation, there are some practical considerations associated with finite-difference methods that favor the use of an explicit filter. Foremost among these considerations is the issue of truncation error. All finite-difference approximations have an associated truncation error that increases with increasing wavenumber. These errors can be quite severe for the smallest resolved scales, and these errors will interfere with the dynamics of the small eddies if no corrective action is taken. Years of experience at CTR with a second-order finite-difference scheme for high Reynolds number LES has repeatedly indicated that truncation errors must be minimized in order to obtain acceptable simulation results. While the potential advantages of explicit filtering are rather clear, there is a significant cost associated with its implementation. In particular, explicit filtering reduces the effective resolution of the simulation compared with that afforded by the mesh. The resolution requirements for LES are usually set by the need to capture

  3. Strong Stability Preserving Explicit Linear Multistep Methods with Variable Step Size

    Hadjimichael, Yiannis


    Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods (of order two and three) with variable step size, and prove their optimality, stability, and convergence. The choice of step size for multistep SSP methods is an interesting problem because the allowable step size depends on the SSP coefficient, which in turn depends on the chosen step sizes. The description of the methods includes an optimal step-size strategy. We prove sharp upper bounds on the allowable step size for explicit SSP linear multistep methods and show the existence of methods with arbitrarily high order of accuracy. The effectiveness of the methods is demonstrated through numerical examples.

  4. The Jeffery-Hamel Flow and Heat Transfer of Nanofluids by Homotopy Perturbation Method and Comparison with Numerical Results

    Pourabdian, Majid; Morad, Mohammad Reza; Javareshkian, Alireza


    This paper considers the influence of nanoparticles on the nonlinear Jeffery-Hamel flow problem. Investigation is performed for three types of nanoparticles namely copper Cu, alumina Al2O3 and titania TiO2 by considering water as a base fluid. The resulting nonlinear governing equations and their associated boundary conditions are solved for both semi-analytical and numerical solutions. The semi-analytical solution is developed by using Homotopy Perturbation Method (HPM) whereas the numerical solution is presented by Runge-Kutta scheme. Dimensionless velocity, temperature, skin friction coefficient and Nusselt number are addressed for the involved pertinent parameters. It is observed that the influence of solid volume fraction of nanoparticles on the heat transfer and fluid flow parameters is more noticeable when compared with the type of nanoparticles. The achieved results reveal that HPM is very effective, convenient and accurate for this problem.

  5. Simulation of near-fault bedrock strong ground-motion field by explicit finite element method

    ZHANG Xiao-zhi; HU Jin-jun; XIE Li-li; WANG Hai-yun


    Based on presumed active fault and corresponding model, this paper predicted the near-fault ground motion filed of a scenario earthquake (Mw=6 3/4 ) in an active fault by the explicit finite element method in combination with the source time function with improved transmitting artificial boundary and with high-frequency vibration contained.The results indicate that the improved artificial boundary is stable in numerical computation and the predicted strong ground motion has a consistent characteristic with the observed motion.

  6. An explicit finite volume element method for solving characteristic level set equation on triangular grids

    Sutthisak Phongthanapanich; Pramote Dechaumphai


    Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.

  7. Time-explicit methods for joint economical and geological risk mitigation in production optimization

    Christiansen, Lasse Hjuler; Capolei, Andrea; Jørgensen, John Bagterp


    Real-life applications of production optimization face challenges of risks related to unpredictable fluctuations in oil prices and sparse geological data. Consequently, operating companies are reluctant to adopt model-based production optimization into their operations. Conventional production...... optimization methods focus on mitigation of geological risks related to the long-term net present value (NPV). A major drawback of such methods is that the time-dependent and exceedingly growing uncertainty of oil prices implies that long-term predictions become highly unreliable. Conventional methods...... of mitigating economical and geological risks. As opposed to conventional strategies that focus on a single long-term objective, TE methods seek to reduce risks and promote returns over the entire reservoir life by optimization of a given ensemble-based geological risk measure over time. By explicit involvement...

  8. a Method to Correct Yield Surface Drift in Soil Plasticity Under Mixed Control and Explicit Integration

    Mattsson, Hans; Axelsson, Kennet; Klisinski, Marek


    When applying an explicit integration algorithm in e.g. soil plasticity, the predicted stress point at the end of an elastoplastic increment of loading might not be situated on the updated current yield surface. This so-called yield surface drift could generally be held under control by using small integration steps. Another possibility, when circumstances might demand larger steps, is to adopt a drift correction method. In this paper, a drift correction method for mixed control in soil plasticity, under drained as well as undrained conditions, is proposed. By simulating triaxial tests in a Constitutive Driver, the capability and efficiency of this correction method, under different choices of implementation, have been analysed. It was concluded that the proposed drift correction method, for quite marginal additional computational cost, was able to correct successfully for yield surface drift giving results in close agreement to those obtained with a very large number of integration steps.

  9. A simple method for finding explicit analytic transition densities of diffusion processes with general diploid selection.

    Song, Yun S; Steinrücken, Matthias


    The transition density function of the Wright-Fisher diffusion describes the evolution of population-wide allele frequencies over time. This function has important practical applications in population genetics, but finding an explicit formula under a general diploid selection model has remained a difficult open problem. In this article, we develop a new computational method to tackle this classic problem. Specifically, our method explicitly finds the eigenvalues and eigenfunctions of the diffusion generator associated with the Wright-Fisher diffusion with recurrent mutation and arbitrary diploid selection, thus allowing one to obtain an accurate spectral representation of the transition density function. Simplicity is one of the appealing features of our approach. Although our derivation involves somewhat advanced mathematical concepts, the resulting algorithm is quite simple and efficient, only involving standard linear algebra. Furthermore, unlike previous approaches based on perturbation, which is applicable only when the population-scaled selection coefficient is small, our method is nonperturbative and is valid for a broad range of parameter values. As a by-product of our work, we obtain the rate of convergence to the stationary distribution under mutation-selection balance.

  10. Analytical exploration of γ-function explicit method for pseudodynamic testing of nonlinear systems

    Shuenn-Yih Chang; Yu-Chi Sung


    It has been well studied that the γ-function explicit method can be effective in providing favorable numerical dissipation for linear elastic systems. However, its performance for nonlinear systems is unclear due to a lack of analytical evaluation techniques. Thus, a novel technique is proposed herein to evaluate its efficiency for application to nonlinear systems by introducing two parameters to describe the stiffness change. As a result, the numerical properties and error propagation characteristics of the γ-function explicit method for the pseudodynamic testing of a nonlinear system are analytically assessed. It is found that the upper stability limit decreases as the step degree of nonlinearity increases; and it increases as the current degree of nonlinearity increases. It is also shown that this integration method provides favorable numerical dissipation not only for linear elastic systems but also for nonlinear systems. Furthermore, error propagation analysis reveals that the numerical dissipation can effectively suppress the severe error propagation of high frequency modes while the low frequency responses are almost unaffected for both linear elastic and nonlinear systems.


    Nuraini Nuraini


    Full Text Available Abstract. Modeling the dynamics of seawater typically uses a shallow water model. The shallow water model is derived from the mass conservation equation and the momentum set into shallow water equations. A two-dimensional shallow water equation alongside the model that is integrated with depth is described in numerical form. This equation can be solved by finite different methods either explicitly or implicitly. In this modeling, the two dimensional shallow water equations are described in discrete form using explicit schemes. Keyword: shallow water equation, finite difference and schema explisit. REFERENSI  1. Bunya, S., Westerink, J. J. dan Yoshimura. 2005. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids. 47: 1451-1468. 2. Kampf Jochen. 2009. Ocean Modelling For Beginners. Springer Heidelberg Dordrecht. London New York. 3. Rezolla, L 2011. Numerical Methods for the Solution of Partial Diferential Equations. Trieste. International Schoolfor Advanced Studies. 4. Natakussumah, K. D., Kusuma, S. B. M., Darmawan, H., Adityawan, B. M. Dan  Farid, M. 2007. Pemodelan Hubungan Hujan dan Aliran Permukaan pada Suatu DAS  dengan Metode Beda Hingga. ITB Sain dan Tek. 39: 97-123. 5. Casulli, V. dan Walters, A. R. 2000. An unstructured grid, three-dimensional model based on the shallow water equations. Int. J. Numer. Meth. Fluids. 32: 331-348. 6. Triatmodjo, B. 2002. Metode Numerik  Beta Offset. Yogyakarta.

  12. Convergence Rates and Explicit Error Bounds of Hill's Method for Spectra of Self-Adjoint Differential Operators

    Tanaka, Ken'ichiro; Murashige, Sunao


    We present the convergence rates and the explicit error bounds of Hill's method, which is a numerical method for computing the spectra of ordinary differential operators with periodic coefficients. This method approximates the operator by a finite dimensional matrix. On the assumption that the operator is selfadjoint, it is shown that, under some conditions, we can obtain the convergence rates of eigenvalues with respect to the dimension and the explicit error bounds. Numerical examples demon...

  13. An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

    Belendez, A., E-mail: a.belendez@ua.e [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Mendez, D.I. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Fernandez, E. [Departamento de Optica, Farmacologia y Anatomia, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Marini, S. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Pascual, I. [Departamento de Optica, Farmacologia y Anatomia, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain)


    The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the 'cubication' of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.

  14. The First-Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations

    Yadong Shang


    Full Text Available This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion-acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.

  15. An Iterative Phase-Space Explicit Discontinuous Galerkin Method for Stellar Radiative Transfer in Extended Atmospheres

    de Almeida, V.F.


    A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicularly to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiative intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiative intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

  16. An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

    de Almeida, Valmor F.


    A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

  17. GPU-acceleration of parallel unconditionally stable group explicit finite difference method

    Parand, K; Hossayni, Sayyed A


    Graphics Processing Units (GPUs) are high performance co-processors originally intended to improve the use and quality of computer graphics applications. Since researchers and practitioners realized the potential of using GPU for general purpose, their application has been extended to other fields out of computer graphics scope. The main objective of this paper is to evaluate the impact of using GPU in solution of the transient diffusion type equation by parallel and stable group explicit finite difference method. To accomplish that, GPU and CPU-based (multi-core) approaches were developed. Moreover, we proposed an optimal synchronization arrangement for its implementation pseudo-code. Also, the interrelation of GPU parallel programming and initializing the algorithm variables was discussed, using numerical experiences. The GPU-approach results are faster than a much expensive parallel 8-thread CPU-based approach results. The GPU, used in this paper, is an ordinary laptop GPU (GT 335M) and is accessible for e...

  18. Thermal Analysis of Ball screw Systems by Explicit Finite Difference Method

    Min, Bog Ki [Hanyang Univ., Seoul (Korea, Republic of); Park, Chun Hong; Chung, Sung Chong [KIMM, Daejeon (Korea, Republic of)


    Friction generated from balls and grooves incurs temperature rise in the ball screw system. Thermal deformation due to the heat degrades positioning accuracy of the feed drive system. To compensate for the thermal error, accurate prediction of the temperature distribution is required first. In this paper, to predict the temperature distribution according to the rotational speed, solid and hollow cylinders are applied for analysis of the ball screw shaft and nut, respectively. Boundary conditions such as the convective heat transfer coefficient, friction torque, and thermal contact conductance (TCC) between balls and grooves are formulated according to operating and fabrication conditions of the ball screw. Explicit FDM (finite difference method) is studied for development of a temperature prediction simulator. Its effectiveness is verified through numerical analysis.

  19. Solution of dynamic contact problems by implicit/explicit methods. Final report

    Salveson, M.W.; Taylor, R.L. [California Univ., Berkeley, CA (United States). Dept. of Civil and Environmental Engineering


    The solution of dynamic contact problems within an explicit finite element program such as the LLNL DYNA programs is addressed in the report. The approach is to represent the solution for the deformation of bodies using the explicit algorithm but to solve the contact part of the problem using an implicit approach. Thus, the contact conditions at the next solution state are considered when computing the acceleration state for each explicit time step.

  20. Connecting free energy surfaces in implicit and explicit solvent: an efficient method to compute conformational and solvation free energies.

    Deng, Nanjie; Zhang, Bin W; Levy, Ronald M


    The ability to accurately model solvent effects on free energy surfaces is important for understanding many biophysical processes including protein folding and misfolding, allosteric transitions, and protein–ligand binding. Although all-atom simulations in explicit solvent can provide an accurate model for biomolecules in solution, explicit solvent simulations are hampered by the slow equilibration on rugged landscapes containing multiple basins separated by barriers. In many cases, implicit solvent models can be used to significantly speed up the conformational sampling; however, implicit solvent simulations do not fully capture the effects of a molecular solvent, and this can lead to loss of accuracy in the estimated free energies. Here we introduce a new approach to compute free energy changes in which the molecular details of explicit solvent simulations are retained while also taking advantage of the speed of the implicit solvent simulations. In this approach, the slow equilibration in explicit solvent, due to the long waiting times before barrier crossing, is avoided by using a thermodynamic cycle which connects the free energy basins in implicit solvent and explicit solvent using a localized decoupling scheme. We test this method by computing conformational free energy differences and solvation free energies of the model system alanine dipeptide in water. The free energy changes between basins in explicit solvent calculated using fully explicit solvent paths agree with the corresponding free energy differences obtained using the implicit/explicit thermodynamic cycle to within 0.3 kcal/mol out of ∼3 kcal/mol at only ∼8% of the computational cost. We note that WHAM methods can be used to further improve the efficiency and accuracy of the implicit/explicit thermodynamic cycle.

  1. A Two-moment Radiation Hydrodynamics Module in Athena Using a Time-explicit Godunov Method

    Skinner, M Aaron


    We describe a module for the Athena code that solves the gray equations of radiation hydrodynamics (RHD), based on the first two moments of the radiative transfer equation. We use a combination of explicit Godunov methods to advance the gas and radiation variables including the non-stiff source terms, and a local implicit method to integrate the stiff source terms. We adopt the M1 closure relation and include all leading source terms. We employ the reduced speed of light approximation (RSLA) with subcycling of the radiation variables in order to reduce computational costs. Our code is dimensionally unsplit in one, two, and three space dimensions and is parallelized using MPI. The streaming and diffusion limits are well-described by the M1 closure model, and our implementation shows excellent behavior for a problem with a concentrated radiation source containing both regimes simultaneously. Our operator-split method is ideally suited for problems with a slowly varying radiation field and dynamical gas flows, i...

  2. A half-explicit, non-split projection method for low Mach number flows.

    Pousin, Jerome G. (National Institute for Applied Sciences, France); Najm, Habib N.; Pebay, Philippe Pierre


    In the context of the direct numerical simulation of low MACH number reacting flows, the aim of this article is to propose a new approach based on the integration of the original differential algebraic (DAE) system of governing equations, without further differentiation. In order to do so, while preserving a possibility of easy parallelization, it is proposed to use a one-step index 2 DAE time-integrator, the Half Explicit Method (HEM). In this context, we recall why the low MACH number approximation belongs to the class of index 2 DAEs and discuss why the pressure can be associated with the constraint. We then focus on a fourth-order HEM scheme, and provide a formulation that makes its implementation more convenient. Practical details about the consistency of initial conditions are discussed, prior to focusing on the implicit solve involved in the method. The method is then evaluated using the Modified KAPS Problem, since it has some of the features of the low MACH number approximation. Numerical results are presented, confirming the above expectations. A brief summary of ongoing efforts is finally provided.

  3. A Two-moment Radiation Hydrodynamics Module in Athena Using a Time-explicit Godunov Method

    Skinner, M. Aaron; Ostriker, Eve C.


    We describe a module for the Athena code that solves the gray equations of radiation hydrodynamics (RHD), based on the first two moments of the radiative transfer equation. We use a combination of explicit Godunov methods to advance the gas and radiation variables including the non-stiff source terms, and a local implicit method to integrate the stiff source terms. We adopt the M 1 closure relation and include all leading source terms to {O}(β τ). We employ the reduced speed of light approximation (RSLA) with subcycling of the radiation variables in order to reduce computational costs. Our code is dimensionally unsplit in one, two, and three space dimensions and is parallelized using MPI. The streaming and diffusion limits are well described by the M 1 closure model, and our implementation shows excellent behavior for a problem with a concentrated radiation source containing both regimes simultaneously. Our operator-split method is ideally suited for problems with a slowly varying radiation field and dynamical gas flows, in which the effect of the RSLA is minimal. We present an analysis of the dispersion relation of RHD linear waves highlighting the conditions of applicability for the RSLA. To demonstrate the accuracy of our method, we utilize a suite of radiation and RHD tests covering a broad range of regimes, including RHD waves, shocks, and equilibria, which show second-order convergence in most cases. As an application, we investigate radiation-driven ejection of a dusty, optically thick shell in the ISM. Finally, we compare the timing of our method with other well-known iterative schemes for the RHD equations. Our code implementation, Hyperion, is suitable for a wide variety of astrophysical applications and will be made freely available on the Web.


    Dong Wang; Steven J. Ruuth


    Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving partial differential equations (PDEs) with terms of different types. While fixed time-step versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit (VSIMEX) linear multistep methods for time-dependent PDEs. Families of order-p, p-step VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers' equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.

  5. Convergence Rates and Explicit Error Bounds of Hill's Method for Spectra of Self-Adjoint Differential Operators

    Tanaka, Ken'ichiro


    We present the convergence rates and the explicit error bounds of Hill's method, which is a numerical method for computing the spectra of ordinary differential operators with periodic coefficients. This method approximates the operator by a finite dimensional matrix. On the assumption that the operator is selfadjoint, it is shown that, under some conditions, we can obtain the convergence rates of eigenvalues with respect to the dimension and the explicit error bounds. Numerical examples demonstrate that we can verify these conditions using Gershgorin's theorem for some real problems. Main theorems are proved using the Dunford integrals which project an eigenvector to the corresponding eigenspace.

  6. Constrained Adiabatic Trajectory Method (CATM): a global integrator for explicitly time-dependent Hamiltonians

    Leclerc, Arnaud; Viennot, David; Killingbeck, John P; 10.1063/1.3673320


    The Constrained Adiabatic Trajectory Method (CATM) is reexamined as an integrator for the Schr\\"odinger equation. An initial discussion places the CATM in the context of the different integrators used in the literature for time-independent or explicitly time-dependent Hamiltonians. The emphasis is put on adiabatic processes and within this adiabatic framework the interdependence between the CATM, the wave operator, the Floquet and the (t,t') theories is presented in detail. Two points are then more particularly analysed and illustrated by a numerical calculation describing the $H_2^+$ ion submitted to a laser pulse. The first point is the ability of the CATM to dilate the Hamiltonian spectrum and thus to make the perturbative treatment of the equations defining the wave function possible, possibly by using a Krylov subspace approach as a complement. The second point is the ability of the CATM to handle extremely complex time-dependencies, such as those which appear when interaction representations are used to...

  7. Constrained adiabatic trajectory method: A global integrator for explicitly time-dependent Hamiltonians

    Leclerc, A.; Jolicard, G.; Viennot, D.; Killingbeck, J. P.


    The constrained adiabatic trajectory method (CATM) is reexamined as an integrator for the Schrödinger equation. An initial discussion places the CATM in the context of the different integrators used in the literature for time-independent or explicitly time-dependent Hamiltonians. The emphasis is put on adiabatic processes and within this adiabatic framework the interdependence between the CATM, the wave operator, the Floquet, and the (t, t') theories is presented in detail. Two points are then more particularly analyzed and illustrated by a numerical calculation describing the H_2^+ ion submitted to a laser pulse. The first point is the ability of the CATM to dilate the Hamiltonian spectrum and thus to make the perturbative treatment of the equations defining the wave function possible, possibly by using a Krylov subspace approach as a complement. The second point is the ability of the CATM to handle extremely complex time-dependencies, such as those which appear when interaction representations are used to integrate the system.

  8. Creating a spatially-explicit index: a method for assessing the global wildfire-water risk

    Robinne, François-Nicolas; Parisien, Marc-André; Flannigan, Mike; Miller, Carol; Bladon, Kevin D.


    The wildfire-water risk (WWR) has been defined as the potential for wildfires to adversely affect water resources that are important for downstream ecosystems and human water needs for adequate water quantity and quality, therefore compromising the security of their water supply. While tools and methods are numerous for watershed-scale risk analysis, the development of a toolbox for the large-scale evaluation of the wildfire risk to water security has only started recently. In order to provide managers and policy-makers with an adequate tool, we implemented a method for the spatial analysis of the global WWR based on the Driving forces-Pressures-States-Impacts-Responses (DPSIR) framework. This framework relies on the cause-and-effect relationships existing between the five categories of the DPSIR chain. As this approach heavily relies on data, we gathered an extensive set of spatial indicators relevant to fire-induced hydrological hazards and water consumption patterns by human and natural communities. When appropriate, we applied a hydrological routing function to our indicators in order to simulate downstream accumulation of potentially harmful material. Each indicator was then assigned a DPSIR category. We collapsed the information in each category using a principal component analysis in order to extract the most relevant pixel-based information provided by each spatial indicator. Finally, we compiled our five categories using an additive indexation process to produce a spatially-explicit index of the WWR. A thorough sensitivity analysis has been performed in order to understand the relationship between the final risk values and the spatial pattern of each category used during the indexation. For comparison purposes, we aggregated index scores by global hydrological regions, or hydrobelts, to get a sense of regional DPSIR specificities. This rather simple method does not necessitate the use of complex physical models and provides a scalable and efficient tool


    Peter G(o)rtz


    Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge Kutta, and Runge-Kutta-Nystrom methods applied to the simple Hamiltonian system p = -vq, q = κp are studied. Some new results in connection with P-stability are pre sented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is the exact solution of a perturbed Hamiltonian system at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.

  10. Proof of concept of a method that assesses the spread of microbial infections with spatially explicit and non-spatially explicit data

    Smith Stephen D


    Full Text Available Abstract Background A method that assesses bacterial spatial dissemination was explored. It measures microbial genotypes (defined by electrophoretic patterns or EP, host, location (farm, interfarm Euclidean distance, and time. Its proof of concept (construct and internal validity was evaluated using a dataset that included 113 Staphylococcus aureus EPs from 1126 bovine milk isolates collected on 23 farms between 1988 and 2005. Results Construct validity was assessed by comparing results based on the interfarm Euclidean distance (a spatially explicit measure and those produced by the (non-spatial interfarm number of isolates reporting the same EP. The distance associated with EP spread correlated with the interfarm number of isolates/EP (r = .59, P r = .72, P r = .87, P Conclusion Findings supported both construct and internal validity. Because 3 EPs explained 12 times more isolates than expected and at least twice as many isolates as other EPs did, false negative results associated with the remaining EPs (those erroneously identified as lacking spatial dispersal when, in fact, they disseminated spatially, if they occurred, seemed to have negligible effects. Spatial analysis of laboratory data may support disease surveillance systems by generating hypotheses on microbial dispersal ability.

  11. Geminal-spanning orbitals make explicitly correlated reduced-scaling coupled-cluster methods robust, yet simple

    Pavošević, Fabijan; Neese, Frank; Valeev, Edward F.


    We present a production implementation of reduced-scaling explicitly correlated (F12) coupled-cluster singles and doubles (CCSD) method based on pair-natural orbitals (PNOs). A key feature is the reformulation of the explicitly correlated terms using geminal-spanning orbitals that greatly reduce the truncation errors of the F12 contribution. For the standard S66 benchmark of weak intermolecular interactions, the cc-pVDZ-F12 PNO CCSD F12 interaction energies reproduce the complete basis set CCSD limit with mean absolute error <0.1 kcal/mol, and at a greatly reduced cost compared to the conventional CCSD F12.

  12. How should we teach everyday skills in dementia? A controlled study comparing implicit and explicit training methods

    Tilborg, I.A.D.A. van; Kessels, R.P.C.; Hulstijn, W.


    Objective: To compare the immediate and delayed effects of implicit and explicit training methods for everyday skills in patients with dementia. Design: Counterbalanced self-controlled cases series. Subjects: Convenience sample of 10 patients with dementia (Mini-Mental State Examination score

  13. How should we teach everyday skills in dementia? A controlled study comparing implicit and explicit training methods

    Tilborg, I.A. Van; Kessels, R.P.C.; Hulstijn, W.


    OBJECTIVE: To compare the immediate and delayed effects of implicit and explicit training methods for everyday skills in patients with dementia. DESIGN: Counterbalanced self-controlled cases series. SUBJECTS: Convenience sample of 10 patients with dementia (Mini-Mental State Examination score

  14. Method matters : Effects of explicit versus implicit social comparisons on activation, behavior, and self-views

    Stapel, DA; Suls, J


    The authors investigated the impact of explicit versus implicit social comparisons. Simply being primed with a superior or inferior standard. (implicit comparison) produced contrast as evidenced by accessibility of self-knowledge (Study 2), intellectual performance (Study 3), and self-ratings (Study


    Sang Dong KIM; Byeong Chun SHIN


    The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to Lu := -u" with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The condition number of the resulting system behaves like O(1/h)where h is the mesh size. We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to O(1).

  16. GPU-Acceleration of Parallel Unconditionally Stable Group Explicit Finite Difference Method

    Parand, K.; Zafarvahedian, Saeed; Hossayni, Sayyed A.


    Graphics Processing Units (GPUs) are high performance co-processors originally intended to improve the use and quality of computer graphics applications. Once, researchers and practitioners noticed the potential of using GPU for general purposes, GPUs applications have been extended from graphics applications to other fields. The main objective of this paper is to evaluate the impact of using GPU in solution of the transient diffusion type equation by parallel and stable group explicit finite...




    The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density.The electric potential equation is discretized by a mixed finite element method.The electron and hole density equations are treated by implicit-explicit multistep finite element methods.The schemes are very efficient.The optimal order error estimates both in time and space are derived.

  18. How should we teach everyday skills in dementia? A controlled study comparing implicit and explicit training methods.

    van Tilborg, Ilse Ada; Kessels, Roy Pc; Hulstijn, Wouter


    To compare the immediate and delayed effects of implicit and explicit training methods for everyday skills in patients with dementia. Counterbalanced self-controlled cases series. Convenience sample of 10 patients with dementia (Mini-Mental State Examination score between 15 and 26) and 16 age- and education-matched controls. Two everyday tasks (using a microwave oven and a coffee machine) that were novel to all participants were trained in five 15-minute sessions. Each participant learned both tasks, one using an implicit learning method (modelling) and the other using an explicit learning method (providing verbal cues). Tasks and conditions were counterbalanced. The participants' performance was videotaped to assess how well the tasks were performed before training, after each training session, and 7-10 days after the final training session. A rater, who was blind to the training method used, scored the number of correctly executed steps by viewing the videotapes. The two training methods were effective in both the patient and healthy control groups, with there being a significant baseline-to-follow-up increase in the number of correctly completed steps (P dementia are able to acquire new skills that are relevant for daily life, showing a similar rate of learning regardless of whether implicit or explicit learning techniques are used.

  19. An explicit four-dimensional variational data assimilation method based on the proper orthogonal decomposition: Theoretics and evaluation

    TIAN XiangJun; XIE ZhengHui


    The proper orthogonal decomposition (POD) method is used to construct a set of basis functions for spanning the ensemble of data in a certain least squares optimal sense. Compared with the singular value decomposition (SVD), the POD basis functions can capture more energy in the forecast ensemble space and can represent its spatial structure and temporal evolution more effectively. After the analysis variables are expressed by a truncated expansion of the POD basis vectors in the ensemble space, the control variables appear explicitly in the cost function, so that the adjoint model, which is used to de-rive the gradient of the cost function with respect to the control variables, is no longer needed. The application of this new technique significantly simplifies the data assimilation process. Several as-similation experiments show that this POD-based explicit four-dimensional variational data assimila-tion method performs much better than the usual ensemble Kalman filter method on both enhancing the assimilation precision and reducing the computation cost. It is also better than the SVD-based ex-plicit four-dimensional assimilation method, especially when the forecast model is not perfect and the forecast error comes from both the noise of the initial filed and the uncertainty of the forecast model.

  20. Parallel Computation on Multicore Processors Using Explicit Form of the Finite Element Method and C++ Standard Libraries

    Rek Václav


    Full Text Available In this paper, the form of modifications of the existing sequential code written in C or C++ programming language for the calculation of various kind of structures using the explicit form of the Finite Element Method (Dynamic Relaxation Method, Explicit Dynamics in the NEXX system is introduced. The NEXX system is the core of engineering software NEXIS, Scia Engineer, RFEM and RENEX. It has the possibilities of multithreaded running, which can now be supported at the level of native C++ programming language using standard libraries. Thanks to the high degree of abstraction that a contemporary C++ programming language provides, a respective library created in this way can be very generalized for other purposes of usage of parallelism in computational mechanics.

  1. Explicit Interaction

    Löwgren, Jonas; Eriksen, Mette Agger; Linde, Per


    as an interpretation of palpability, comprising usability as well as patient empowerment and socially performative issues. We present a prototype environment for video recording during physiotherapeutical consultation which illustrates our current thoughts on explicit interaction and serves as material for further......We report an ongoing study of palpable computing to support surgical rehabilitation, in the general field of interaction design for ubiquitous computing. Through explorative design, fieldwork and participatory design techniques, we explore the design principle of explicit interaction...

  2. Coupling Chemical Kinetics and Flashes in Reactive, Thermal and Compositional Reservoir Simulation

    Kristensen, Morten Rode; Gerritsen, Margot G.; Thomsen, Per Grove;


    of convergence and error test failures by more than 50% compared to direct integration without the new algorithm. To facilitate the algorithmic development we construct a virtual kinetic cell model. We use implicit one-step ESDIRK (Explicit Singly Diagonal Implicit Runge-Kutta) methods for integration...... of the kinetics. The kinetic cell model serves both as a tool for the development and testing of tailored solvers as well as a testbed for studying the interactions between chemical kinetics and phase behavior. A comparison between a Kvalue correlation based approach and a more rigorous equation of state based...

  3. Explicit FE wrinkling simulation and method to catch critical bifurcation point in tube bending process

    YANG He; LI Heng; ZHAN Mei; GU Rui-jie


    The wrinkling has become the main defect in the thin-walled tube NC bending process. In the study, a dynamic explicit FE model for aluminum alloy thin-walled tube NC bending process is developed to predict the wrinkling by using FE code ABAQUS/Explicit. Attention was paid to the influences of mass scaling, loading rate scaling, mesh density and element type on accurate wrinkling prediction. So the wrinkling modes and mechanism are revealed based on the reliable FE model. Then a two step strategy is proposed to capture the critical bifurcation point for the optimal design process. The results show: 1) The boundary conditions determine the tube materials response greatly so that the frequency analysis is meaningless to the simulation. It is the contact conditions that make the effect of the mass scaling and loading rate less significant.2) There are two wrinkling modes in the tube bending process. One refers to that local ripples occur initially in the straight regions contacted with wiper die and mandrel; the other refers to that local wrinkles occur in the curved regions due to the relative slipping between tube and clamp die. 3) Both the difference of the in-plane compressive stresses and the relative slipping distance are chosen to be the quantitative indexes to represent the critical point and wrinkling tendency. The experiment of aluminum alloy (5052 O) tube bending was carried out to verify whether the above wrinkle modes exist and the indexes proposed are reasonable to catch the critical bifurcation point. The results may help better understanding of the wrinkling mechanism and the process optimization of the tube bending.

  4. Methods used to parameterize the spatially-explicit components of a state-and-transition simulation model

    Rachel R. Sleeter


    Full Text Available Spatially-explicit state-and-transition simulation models of land use and land cover (LULC increase our ability to assess regional landscape characteristics and associated carbon dynamics across multiple scenarios. By characterizing appropriate spatial attributes such as forest age and land-use distribution, a state-and-transition model can more effectively simulate the pattern and spread of LULC changes. This manuscript describes the methods and input parameters of the Land Use and Carbon Scenario Simulator (LUCAS, a customized state-and-transition simulation model utilized to assess the relative impacts of LULC on carbon stocks for the conterminous U.S. The methods and input parameters are spatially explicit and describe initial conditions (strata, state classes and forest age, spatial multipliers, and carbon stock density. Initial conditions were derived from harmonization of multi-temporal data characterizing changes in land use as well as land cover. Harmonization combines numerous national-level datasets through a cell-based data fusion process to generate maps of primary LULC categories. Forest age was parameterized using data from the North American Carbon Program and spatially-explicit maps showing the locations of past disturbances (i.e. wildfire and harvest. Spatial multipliers were developed to spatially constrain the location of future LULC transitions. Based on distance-decay theory, maps were generated to guide the placement of changes related to forest harvest, agricultural intensification/extensification, and urbanization. We analyze the spatially-explicit input parameters with a sensitivity analysis, by showing how LUCAS responds to variations in the model input. This manuscript uses Mediterranean California as a regional subset to highlight local to regional aspects of land change, which demonstrates the utility of LUCAS at many scales and applications.

  5. Methods used to parameterize the spatially-explicit components of a state-and-transition simulation model

    Sleeter, Rachel; Acevedo, William; Soulard, Christopher E.; Sleeter, Benjamin M.


    Spatially-explicit state-and-transition simulation models of land use and land cover (LULC) increase our ability to assess regional landscape characteristics and associated carbon dynamics across multiple scenarios. By characterizing appropriate spatial attributes such as forest age and land-use distribution, a state-and-transition model can more effectively simulate the pattern and spread of LULC changes. This manuscript describes the methods and input parameters of the Land Use and Carbon Scenario Simulator (LUCAS), a customized state-and-transition simulation model utilized to assess the relative impacts of LULC on carbon stocks for the conterminous U.S. The methods and input parameters are spatially explicit and describe initial conditions (strata, state classes and forest age), spatial multipliers, and carbon stock density. Initial conditions were derived from harmonization of multi-temporal data characterizing changes in land use as well as land cover. Harmonization combines numerous national-level datasets through a cell-based data fusion process to generate maps of primary LULC categories. Forest age was parameterized using data from the North American Carbon Program and spatially-explicit maps showing the locations of past disturbances (i.e. wildfire and harvest). Spatial multipliers were developed to spatially constrain the location of future LULC transitions. Based on distance-decay theory, maps were generated to guide the placement of changes related to forest harvest, agricultural intensification/extensification, and urbanization. We analyze the spatially-explicit input parameters with a sensitivity analysis, by showing how LUCAS responds to variations in the model input. This manuscript uses Mediterranean California as a regional subset to highlight local to regional aspects of land change, which demonstrates the utility of LUCAS at many scales and applications.

  6. The Influence of Alloying and Processing on the Microstructure and Properties of Beta-NiAl.


    height, the static Young-Laplace equation is solved using an adaptive fifth order Runge-Kutta method and the shooting technique to generate- all...for melt zone height and PL using the shooting method with Runge-Kutta fifth order integration. (a) Profiles satisfying the static Young-Laplace...has been reported that the toughnes’ of e specimens oriente-d with (100) as th-. cra :’k, plane is nearly twvice the toughness of sp:.-imen s having

  7. Molecular dynamics simulations of double-stranded DNA in an explicit solvent model with the zero-dipole summation method.

    Takamasa Arakawa

    Full Text Available Molecular dynamics (MD simulations of a double-stranded DNA with explicit water and small ions were performed with the zero-dipole summation (ZD method, which was recently developed as one of the non-Ewald methods. Double-stranded DNA is highly charged and polar, with phosphate groups in its backbone and their counterions, and thus precise treatment for the long-range electrostatic interactions is always required to maintain the stable and native double-stranded form. A simple truncation method deforms it profoundly. On the contrary, the ZD method, which considers the neutralities of charges and dipoles in a truncated subset, well reproduced the electrostatic energies of the DNA system calculated by the Ewald method. The MD simulations using the ZD method provided a stable DNA system, with similar structures and dynamic properties to those produced by the conventional Particle mesh Ewald method.

  8. Exact and explicit solutions to the (3+1)-dimensional Jimbo-Miwa equation via the Exp-function method

    Ozis, Turgut [Department of Mathematics, Ege University, 35100 Bornova, Izmir (Turkey)], E-mail:; Aslan, Ismail [Department of Mathematics, Izmir Institute of Technology, 35430 Urla, Izmir (Turkey)


    In this Letter, the Exp-function method, with the aid of a symbolic computation system such as Mathematica, is applied to the (3+1)-dimensional Jimbo-Miwa equation to show its effectiveness and reliability. Exact and explicit generalized solitary solutions are obtained in more general forms. The free parameters can be determined by initial or boundary conditions. Being less restrictive and concise, the method can be applied to many high-dimensional nonlinear evolution equations having wide applications in applied physical sciences.

  9. Second-order explicit finite-difference methods for transient-flow analysis

    Chaudhry, M. H.; Hussaini, M. Y.


    Three second-order accurate numerical methods - MacCormack's method, Lambda scheme and Gabutti scheme - are introduced to solve the quasi-linear, hyperbolic partial differential equations describing transient flows in closed conduits. The details of these methods and the treatment of boundary conditions are presented and the results computed by using these methods for a typical piping system are compared. It is shown that for the same accuracy, second-order methods require considerably lesser number of computational nodes and computer time as compared to those required by the first-order methods.

  10. Explicit formula of finite difference method to estimate human peripheral tissue temperatures during exposure to severe cold stress.

    Khanday, M A; Hussain, Fida


    During cold exposure, peripheral tissues undergo vasoconstriction to minimize heat loss to preserve the maintenance of a normal core temperature. However, vasoconstricted tissues exposed to cold temperatures are susceptible to freezing and frostbite-related tissue damage. Therefore, it is imperative to establish a mathematical model for the estimation of tissue necrosis due to cold stress. To this end, an explicit formula of finite difference method has been used to obtain the solution of Pennes' bio-heat equation with appropriate boundary conditions to estimate the temperature profiles of dermal and subdermal layers when exposed to severe cold temperatures. The discrete values of nodal temperature were calculated at the interfaces of skin and subcutaneous tissues with respect to the atmospheric temperatures of 25 °C, 20 °C, 15 °C, 5 °C, -5 °C and -10 °C. The results obtained were used to identify the scenarios under which various degrees of frostbite occur on the surface of skin as well as the dermal and subdermal areas. The explicit formula of finite difference method proposed in this model provides more accurate predictions as compared to other numerical methods. This model of predicting tissue temperatures provides researchers with a more accurate prediction of peripheral tissue temperature and, hence, the susceptibility to frostbite during severe cold exposure.

  11. A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations

    Meyer, Chad D.; Balsara, Dinshaw S. [Physics Department, Univ. of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556 (United States); Aslam, Tariq D. [WX-9 Group, Los Alamos National Laboratory, MS P952, Los Alamos, NM 87545 (United States)


    Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s{sup 2} times larger than a single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems – a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very

  12. Analysis of High Order Difference Methods for Multiscale Complex Compressible Flows

    Sjoegreen, Bjoern; Yee, H. C.; Tang, Harry (Technical Monitor)


    Accurate numerical simulations of complex multiscale compressible viscous flows, especially high speed turbulence combustion and acoustics, demand high order schemes with adaptive numerical dissipation controls. Standard high resolution shock-capturing methods are too dissipative to capture the small scales and/or long-time wave propagations without extreme grid refinements and small time steps. An integrated approach for the control of numerical dissipation in high order schemes with incremental studies was initiated. Here we further refine the analysis on, and improve the understanding of the adaptive numerical dissipation control strategy. Basically, the development of these schemes focuses on high order nondissipative schemes and takes advantage of the progress that has been made for the last 30 years in numerical methods for conservation laws, such as techniques for imposing boundary conditions, techniques for stability at shock waves, and techniques for stable and accurate long-time integration. We concentrate on high order centered spatial discretizations and a fourth-order Runge-Kutta temporal discretizations as the base scheme. Near the bound-aries, the base scheme has stable boundary difference operators. To further enhance stability, the split form of the inviscid flux derivatives is frequently used for smooth flow problems. To enhance nonlinear stability, linear high order numerical dissipations are employed away from discontinuities, and nonlinear filters are employed after each time step in order to suppress spurious oscillations near discontinuities to minimize the smearing of turbulent fluctuations. Although these schemes are built from many components, each of which is well-known, it is not entirely obvious how the different components be best connected. For example, the nonlinear filter could instead have been built into the spatial discretization, so that it would have been activated at each stage in the Runge-Kutta time stepping. We could think

  13. Development and Comparison of Numerical Fluxes for LWDG Methods

    Jianxian Qiu


    The discontinuous Galerkin (DG) or local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection-diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. The Lax-Wendroff time discretization procedure is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge-Kutta time discretizations. In this paper, we develop fluxes for the method of DG with Lax-Wendroff time discretization procedure (LWDG) based on different numerical fluxes for finite volume or finite difference schemes, including the first-order monotone fluxes such as the Lax-Friedrichs flux, Godunov flux, the Engquist-Osher flux etc. And the second-order TVD fluxes. We systematically investigate the performance of the LWDG methods based on these differ-ent numerical fluxes for convection terms with the objective of obtaining better perfor-mance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, ac-curacy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.

  14. Wavenumber Explicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation


    1939 of Lecture Notes in Mathematics, Springer-Verlag, Berlin , 2008, pp. x+235. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26...July 1, 2006, Edited by Boffi and Lucia Gastaldi. [12] X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave...Mathematics, Springer-Verlag, Berlin , 1995. Reprint of the sixth (1980) edition. [25] J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and

  15. Using case method to explicitly teach formative assessment in preservice teacher science education

    Bentz, Amy Elizabeth

    The process of formative assessment improves student understanding; however, the topic of formative assessment in preservice education has been severely neglected. Since a major goal of teacher education is to create reflective teaching professionals, preservice teachers should be provided an opportunity to critically reflect on the use of formative assessment in the classroom. Case method is an instructional methodology that allows learners to engage in and reflect on real-world situations. Case based pedagogy can play an important role in enhancing preservice teachers' ability to reflect on teaching and learning by encouraging alternative ways of thinking about assessment. Although the literature on formative assessment and case methodology are extensive, using case method to explore the formative assessment process is, at best, sparse. The purpose of this study is to answer the following research questions: To what extent does the implementation of formative assessment cases in methods instruction influence preservice elementary science teachers' knowledge of formative assessment? What descriptive characteristics change between the preservice teachers' pre-case and post-case written reflection that would demonstrate learning had occurred? To investigate these questions, preservice teachers in an elementary methods course were asked to reflect on and discuss five cases. Pre/post-case data was analyzed. Results indicate that the preservice teachers modified their ideas to reflect the themes that were represented within the cases and modified their reflections to include specific ideas or examples taken directly from the case discussions. Comparing pre- and post-case reflections, the data supports a noted change in how the preservice teachers interpreted the case content. The preservice teachers began to evaluate the case content, question the lack of formative assessment concepts and strategies within the case, and apply formative assessment concepts and

  16. An ESDIRK Method with Sensitivity Analysis Capabilities

    Kristensen, Morten Rode; Jørgensen, John Bagterp; Thomsen, Per Grove


    A new algorithm for numerical sensitivity analysis of ordinary differential equations (ODEs) is presented. The underlying ODE solver belongs to the Runge-Kutta family. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure...

  17. Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs

    Niemeyer, Kyle E


    The chemical kinetics ODEs arising from operator-split reactive-flow simulations were solved on GPUs using explicit integration algorithms. Nonstiff chemical kinetics of a hydrogen oxidation mechanism (9 species and 38 irreversible reactions) were computed using the explicit fifth-order Runge-Kutta-Cash-Karp method, and the GPU-accelerated version performed faster than single- and six-core CPU versions by factors of 126 and 25, respectively, for 524,288 ODEs. Moderately stiff kinetics, represented with mechanisms for hydrogen/carbon-monoxide (13 species and 54 irreversible reactions) and methane (53 species and 634 irreversible reactions) oxidation, were computed using the stabilized explicit second-order Runge-Kutta-Chebyshev (RKC) algorithm. The GPU-based RKC implementation demonstrated an increase in performance of nearly 59 and 10 times, for problem sizes consisting of 262,144 ODEs and larger, than the single- and six-core CPU-based RKC algorithms using the hydrogen/carbon-monoxide mechanism. With the met...

  18. A Collocation Method for Numerical Solutions of Coupled Burgers' Equations

    Mittal, R. C.; Tripathi, A.


    In this paper, we propose a collocation-based numerical scheme to obtain approximate solutions of coupled Burgers' equations. The scheme employs collocation of modified cubic B-spline functions. We have used modified cubic B-spline functions for unknown dependent variables u, v, and their derivatives w.r.t. space variable x. Collocation forms of the partial differential equations result in systems of first-order ordinary differential equations (ODEs). In this scheme, we did not use any transformation or linearization method to handle nonlinearity. The obtained system of ODEs has been solved by strong stability preserving the Runge-Kutta method. The proposed scheme needs less storage space and execution time. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in earlier studies. The scheme is simple as well as easy to implement. The scheme provides approximate solutions not only at the grid points, but also at any point in the solution range.

  19. Efficient computation method for two-dimensional nonlinear waves


    The theory and simulation of fully-nonlinear waves in a truncated two-dimensional wave tank in time domain are presented. A piston-type wave-maker is used to generate gravity waves into the tank field in finite water depth. A damping zone is added in front of the wave-maker which makes it become one kind of absorbing wave-maker and ensures the prescribed Neumann condition. The efficiency of nmerical tank is further enhanced by installation of a sponge layer beach (SLB) in front of downtank to absorb longer weak waves that leak through the entire wave train front. Assume potential flow, the space- periodic irrotational surface waves can be represented by mixed Euler- Lagrange particles. Solving the integral equation at each time step for new normal velocities, the instantaneous free surface is integrated following time history by use of fourth-order Runge- Kutta method. The double node technique is used to deal with geometric discontinuity at the wave- body intersections. Several precise smoothing methods have been introduced to treat surface point with high curvature. No saw-tooth like instability is observed during the total simulation.The advantage of proposed wave tank has been verified by comparing with linear theoretical solution and other nonlinear results, excellent agreement in the whole range of frequencies of interest has been obtained.

  20. Kinematic source model for simulation of near-fault ground motion field using explicit finite element method

    Zhang Xiaozhi; Hu Jinjun; Xie Lili; Wang Haiyun


    This paper briefly reviews the characteristics and major processes of the explicit finite element method in modeling the near-fault ground motion field. The emphasis is on the finite element-related problems in the finite fault source modeling. A modified kinematic source model is presented, in which vibration with some high frequency components is introduced into the traditional slip time function to ensure that the source and ground motion include sufficient high frequency components. The model presented is verified through a simple modeling example. It is shown that the predicted near-fault ground motion field exhibits similar characteristics to those observed in strong motion records, such as the hanging wall effect, vertical effect, fling step effect and velocity pulse effect, etc.

  1. An almost symmetric Strang splitting scheme for the construction of high order composition methods.

    Einkemmer, Lukas; Ostermann, Alexander


    In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure cannot be computed exactly. Instead, we insert a well-chosen state [Formula: see text] into the corresponding nonlinearity [Formula: see text], which results in a linear term [Formula: see text] whose exact flow can be determined efficiently. Therefore, in the spirit of splitting methods, it is still possible for the numerical simulation to satisfy certain properties of the exact flow. However, Strang splitting is no longer symmetric (even though it is still a second order method) and thus high order composition methods are not easily attainable. We will show that an iterated Strang splitting scheme can be constructed which yields a method that is symmetric up to a given order. This method can then be used to attain high order composition schemes. We will illustrate our theoretical results, up to order six, by conducting numerical experiments for a charged particle in an inhomogeneous electric field, a post-Newtonian computation in celestial mechanics, and a nonlinear population model and show that the methods constructed yield superior efficiency as compared to Strang splitting. For the first example we also perform a comparison with the standard fourth order Runge-Kutta methods and find significant gains in efficiency as well better conservation properties.

  2. A Study of the Effect of Infilled Brick Walls on Behavior of Eccentrically Braced Frames Using Explicit Finite Elements Method

    Amir S. Daryan


    Full Text Available Problem statement: Eccentrically Braced Frames (EBFs are usually infilled by masonry walls, but in common design, the stiffness and lateral resistance of these walls is ignored. Considering the results of carried out tests and studies, it seems that infilled masonry walls have a significant influence on the stiffness and the strength of EBFs. Since experimental test of total frame with infilled brick wall is a very expensive and time consuming process, proper numerical models which can precisely simulate the behavior of EBFs considering the effects of infilled brick wall are necessary. Approach: In this study, a proper model is made using explicit finite elements method to study the behavior of EBFs with infilled masonry wall. Because of complicated mechanical and geometrical properties of masonry walls and also because of the interaction between steel frame and masonry wall, this model is not easy to obtain. To ensure the ability of the model to precisely simulate the behavior of an EBF with infilled brick wall, initial models were made and the problems were solved comparing the results of experimental test and the results of these initial models. Firstly, material models and some basic principles of explicit finite element algorithm are used and three initial models were made: a model of a brick wall without eccentrically braced steel frame, a model of an EBF without infilled brick wall and finally a model of an EBF with infilled brick wall. Results: Using these three initial models, constitutive model for masonry and steel material and also the proper elements for modeling the behavior of mortar is obtained. Studies showed that good prediction of the behavior of a system consisting of EBF and masonry wall is possible, by minimizing the kinematical energy and using a special time scaling of explicit finite element model. Conclusion: After verifying the finite element models, the influence of masonry infilled wall on the behavior of

  3. A finite volume method for fluctuating hydrodynamics of simple fluids

    Narayanan, Kiran; Samtaney, Ravi; Moran, Brian


    Fluctuating hydrodynamics accounts for stochastic effects that arise at mesoscopic and macroscopic scales. We present a finite volume method for numerical solutions of the fluctuating compressible Navier Stokes equations. Case studies for simple fluids are demonstrated via the use of two different equations of state (EOS) : a perfect gas EOS, and a Lennard-Jones EOS for liquid argon developed by Johnson et al. (Mol. Phys. 1993). We extend the fourth order conservative finite volume scheme originally developed by McCorquodale and Colella (Comm. in App. Math. & Comput. Sci. 2011), to evaluate the deterministic and stochastic fluxes. The expressions for the cell-centered discretizations of the stochastic shear stress and stochastic heat flux are adopted from Espanol, P (Physica A. 1998), where the discretizations were shown to satisfy the fluctuation-dissipation theorem. A third order Runge-Kutta scheme with weights proposed by Delong et al. (Phy. Rev. E. 2013) is used for the numerical time integration. Accuracy of the proposed scheme will be demonstrated. Comparisons of the numerical solution against theory for a perfect gas as well as liquid argon will be presented. Regularizations of the stochastic fluxes in the limit of zero mesh sizes will be discussed. Supported by KAUST Baseline Research Funds.

  4. Parallel Gridless Method for Numerical Simulation of Reactive Flows%化学反应流模拟的并行无网格方法

    吴伟; 许厚谦; 王亮; 薛锐


    To enlarge the computation scale of numerical simulation of complex reactive flows,the parallel gridless method coupled with finite rate chemical model,was studied based on Message Passing Interface.The fluid dynamics process was described by Euler equation with chemical source in 2-D axisymmetric coordinate,and the numerical method was based on least-square gridless method.The inviscid flux was calculated by multi-component HLLC (Harten-Lax-van Leer-Contact)scheme,and the multistage Runge-Kutta algorithm was used to advance the equations in time.The flows of shock-induced combustion and the supersonic proj ectile-induced oblique detonation were simulated using 2-8 processes respectively.The results show well agreement with the shadowgraph and other numerical results,and the parallel efficiency is accredited.It’s effective to employ this parallel gridless method in the simulation of supersonic reactive flows in engineering applications.%为进一步扩大无网格方法在复杂化学反应流场模拟中的计算规模,基于 MPI(Message Passing Interface)并行环境,发展了耦合有限速率反应模型的并行无网格算法,其流体动力学采用包含反应源项的二维轴对称 Euler方程建模,对流通量采用多组分 HLLC(Harten-Lax-van Leer-contact)格式计算,时间项运用4阶 Runge-Kutta 法显式推进。分别采用2~8个进程对激波诱导燃烧流场以及高速运动弹丸诱导斜爆轰流场进行了数值模拟,其结果同实验以及网格方法获得的结果吻合较好,并且具有较理想的并行效率,验证了其在复杂化学反应流大规模计算中应用的正确性和有效性。

  5. Development and Application of Explicitly Correlated Wave Function Based Methods for the Investigation of Optical Properties of Semiconductor Nanomaterials

    Elward, Jennifer Mary

    Semiconductor nanoparticles, or quantum dots (QDs), are well known to have very unique optical and electronic properties. These properties can be controlled and tailored as a function of several influential factors, including but not limited to the particle size and shape, effect of composition and heterojunction as well as the effect of ligand on the particle surface. This customizable nature leads to extensive experimental and theoretical research on the capabilities of these quantum dots for many application purposes. However, in order to be able to understand and thus further the development of these materials, one must first understand the fundamental interaction within these nanoparticles. In this thesis, I have developed a theoretical method which is called electron-hole explicitly correlated Hartee-Fock (eh-XCHF). It is a variational method for solving the electron-hole Schrodinger equation and has been used in this work to study electron-hole interaction in semiconductor quantum dots. The method was benchmarked with respect to a parabolic quantum dot system, and ground state energy and electron-hole recombination probability were computed. Both of these properties were found to be in good agreement with expected results. Upon successful benchmarking, I have applied the eh-XCHF method to study optical properties of several quantum dot systems including the effect of dot size on exciton binding energy and recombination probability in a CdSe quantum dot, the effect of shape on a CdSe quantum dot, the effect of heterojunction on a CdSe/ZnS quantum dot and the effect of quantum dot-biomolecule interaction within a CdSe-firefly Luciferase protein conjugate system. As metrics for assessing the effect of these influencers on the electron-hole interaction, the exciton binding energy, electron-hole recombination probability and the average electron-hole separation distance have been computed. These excitonic properties have been found to be strongly infuenced by the

  6. Elastic wave propagation in variable media using a discontinuous Galerkin method.

    Ober, Curtis Curry; Smith, Thomas Michael; Collis, Samuel Scott; Overfelt, James Robert; Schwaiger, Hans


    Motivated by the needs of seismic inversion and building on our prior experience for fluid-dynamics systems, we present a high-order discontinuous Galerkin (DG) Runge-Kutta method applied to isotropic, linearized elasto-dynamics. Unlike other DG methods recently presented in the literature, our method allows for inhomogeneous material variations within each element that enables representation of realistic earth models - a feature critical for future use in seismic inversion. Likewise, our method supports curved elements and hybrid meshes that include both simplicial and nonsimplicial elements. We demonstrate the capabilities of this method through a series of numerical experiments including hybrid mesh discretizations of the Marmousi2 model as well as a modified Marmousi2 model with a oscillatory ocean bottom that is exactly captured by our discretization. A discontinuous Galerkin method for solving the equations of linear isotropic elasticity has been presented. The formulation is designed to accommodate variation of media parameters within elements, curved elements and unstructured heterogeneous meshes. We have demonstrated that each of these important features of the formulation can produce results that are significantly different from formulations that do not possess these capabilities suggesting that each of these capabilities may be important for effective full waveform inversion of elastic medium.

  7. Solution of deterministic-stochastic epidemic models by dynamical Monte Carlo method

    Aièllo, O. E.; Haas, V. J.; daSilva, M. A. A.; Caliri, A.


    This work is concerned with dynamical Monte Carlo (MC) method and its application to models originally formulated in a continuous-deterministic approach. Specifically, a susceptible-infected-removed-susceptible (SIRS) model is used in order to analyze aspects of the dynamical MC algorithm and achieve its applications in epidemic contexts. We first examine two known approaches to the dynamical interpretation of the MC method and follow with the application of one of them in the SIRS model. The working method chosen is based on the Poisson process where hierarchy of events, properly calculated waiting time between events, and independence of the events simulated, are the basic requirements. To verify the consistence of the method, some preliminary MC results are compared against exact steady-state solutions and other general numerical results (provided by Runge-Kutta method): good agreement is found. Finally, a space-dependent extension of the SIRS model is introduced and treated by MC. The results are interpreted under and in accordance with aspects of the herd-immunity concept.

  8. Properties-preserving high order numerical methods for a kinetic eikonal equation

    Luo, Songting; Payne, Nicholas


    For the BGK (Bhatnagar-Gross-Krook) equation in the large scale hyperbolic limit, the density of particles can be transformed as the Hopf-Cole transformation, where the phase function converges uniformly to the viscosity solution of an effective Hamilton-Jacobi equation, referred to as the kinetic eikonal equation. In this work, we present efficient high order finite difference methods for numerically solving the kinetic eikonal equation. The methods are based on monotone schemes such as the Godunov scheme. High order weighted essentially non-oscillatory techniques and Runge-Kutta procedures are used to obtain high order accuracy in both space and time. The effective Hamiltonian is determined implicitly by a nonlinear equation given as integrals with respect to the velocity variable. Newton's method is applied to solve the nonlinear equation, where integrals with respect to the velocity variable are evaluated either by a Gauss quadrature formula or as expansions with respect to moments of the Maxwellian. The methods are designed such that several key properties such as the positivity of the viscosity solution and the positivity of the effective Hamiltonian are preserved. Numerical experiments are presented to demonstrate the effectiveness of the methods.

  9. Meshless Local Discontinuous Petrov-Galerkin Method with Application to Blasting Problems

    QIANG Hongfu; GAO Weiran


    A meshless local discontinuous Petrov-Galerkin (MLDPG)method based on the local symmetric weak form(LSWF)is presented with the application to blasting problems.The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin(RKDG)method.The solutions are reproduced in a set of overlapped spherical sub-domains.and the test functions are employed from a partition of unlty of the lpeal basis functions.There is no need of any traditional nonoverlapping mesh either for lpeal approximation purpose or for Galerkin integration purpose in the presented method.The resulting MLDPG method is a meshless.stable.high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method (MM),and it can handle the problems with extremely complicated physics and geometries easily.Three numerical exampies of the one-dimensional Sod shock-tube problem.the blast-wave problem and the Woodward-Cpiella interacting shock wave problem are given.All the numerical results are in good agreement with the closed solutions.The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.

  10. Development of efficient time-evolution method based on three-term recurrence relation

    Akama, Tomoko, E-mail:; Kobayashi, Osamu; Nanbu, Shinkoh, E-mail: [Department of Materials and Life Science, Faculty of Science and Technology, Sophia University, Tokyo 102-8554 (Japan)


    The advantage of the real-time (RT) propagation method is a direct solution of the time-dependent Schrödinger equation which describes frequency properties as well as all dynamics of a molecular system composed of electrons and nuclei in quantum physics and chemistry. Its applications have been limited by computational feasibility, as the evaluation of the time-evolution operator is computationally demanding. In this article, a new efficient time-evolution method based on the three-term recurrence relation (3TRR) was proposed to reduce the time-consuming numerical procedure. The basic formula of this approach was derived by introducing a transformation of the operator using the arcsine function. Since this operator transformation causes transformation of time, we derived the relation between original and transformed time. The formula was adapted to assess the performance of the RT time-dependent Hartree-Fock (RT-TDHF) method and the time-dependent density functional theory. Compared to the commonly used fourth-order Runge-Kutta method, our new approach decreased computational time of the RT-TDHF calculation by about factor of four, showing the 3TRR formula to be an efficient time-evolution method for reducing computational cost.

  11. Petrov-Galerkin Method for the Coupled Schrödinger-KdV Equation

    M. S. Ismail


    Full Text Available Petrov-Galerkin method is used to derive a numerical scheme for the coupled Schrödinger-KdV (SKdV equations, where we have used the cubic B-splines as a test functions and a linear B-splines as a trial functions. Product approximation technique is used to deal with the nonlinear terms. An implicit midpoint rule and the Runge-Kutta method of fourth-order (RK4 are used to discretize in time. A block nonlinear pentadiagonal system is obtained. We solve this system by the fixed point method. The resulting scheme has a fourth-order accuracy in space direction and second-order in time direction in case of the implicit midpoint rule and it is unconditionally stable by von Neumann method. Using the RK4 method the scheme will be linear and fourth-order in time and space directions, and it is also conditionally stable. The exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness and the efficiency of the proposed schemes.

  12. Numerical Treatment of the Model for HIV Infection of CD4+T Cells by Using Multistep Laplace Adomian Decomposition Method

    Nurettin Doğan


    Full Text Available A new method for approximate analytic series solution called multistep Laplace Adomian Decomposition Method (MLADM has been proposed for solving the model for HIV infection of CD4+T cells. The proposed method is modification of the classical Laplace Adomian Decomposition Method (LADM with multistep approach. Fourth-order Runge-Kutta method (RK4 is used to evaluate the effectiveness of the proposed algorithm. When we do not know the exact solution of a given problem, generally we use the RK4 method for comparison since it is widely used and accepted. Comparison of the results with RK4 method is confirmed that MLADM performs with very high accuracy. Results show that MLADM is a very promising method for obtaining approximate solutions to the model for HIV infection of CD4+T cells. Some plots and tables are presented to show the reliability and simplicity of the methods. All computations have been made with the aid of a computer code written in Mathematica 7.

  13. Computation of Aerodynamic Noise Radiated from Ducted Tail Rotor Using Boundary Element Method

    Yunpeng Ma


    Full Text Available A detailed aerodynamic performance of a ducted tail rotor in hover has been numerically studied using CFD technique. The general governing equations of turbulent flow around ducted tail rotor are given and directly solved by using finite volume discretization and Runge-Kutta time integration. The calculations of the lift characteristics of the ducted tail rotor can be obtained. In order to predict the aerodynamic noise, a hybrid method combining computational aeroacoustic with boundary element method (BEM has been proposed. The computational steps include the following: firstly, the unsteady flow around rotor is calculated using the CFD method to get the noise source information; secondly, the radiate sound pressure is calculated using the acoustic analogy Curle equation in the frequency domain; lastly, the scattering effect of the duct wall on the propagation of the sound wave is presented using an acoustic thin-body BEM. The aerodynamic results and the calculated sound pressure levels are compared with the known technique for validation. The sound pressure directivity and scattering effect are shown to demonstrate the validity and applicability of the method.

  14. Verification of higher-order discontinuous Galerkin method for hexahedral elements

    Özdemir, Hüseyin; Hagmeijer, Rob; Hoeijmakers, Hendrik Willem Marie


    A high-order implementation of the Discontinuous Galerkin ( DG) method is presented for solving the three-dimensional Linearized Euler Equations on an unstructured hexahedral grid. The method is based on a quadrature free implementation and the high-order accuracy is obtained by employing higher-degree polynomials as basis functions. The present implementation is up to fourth-order accurate in space. For the time discretization a four-stage Runge-Kutta scheme is used which is fourth-order accurate. Non-reflecting boundary conditions are implemented at the boundaries of the computational domain.The method is verified for the case of the convection of a 1D compact acoustic disturbance. The numerical results show that the rate of convergence of the method is of order p+1 in the mesh size, with p the order of the basis functions. This observation is in agreement with analysis presented in the literature. To cite this article: H. Özdemir et al., C. R. Mecanique 333 (2005).

  15. Accelerated solution of non-linear flow problems using Chebyshev iteration polynomial based RK recursions

    Lorber, A.A.; Carey, G.F.; Bova, S.W.; Harle, C.H. [Univ. of Texas, Austin, TX (United States)


    The connection between the solution of linear systems of equations by iterative methods and explicit time stepping techniques is used to accelerate to steady state the solution of ODE systems arising from discretized PDEs which may involve either physical or artificial transient terms. Specifically, a class of Runge-Kutta (RK) time integration schemes with extended stability domains has been used to develop recursion formulas which lead to accelerated iterative performance. The coefficients for the RK schemes are chosen based on the theory of Chebyshev iteration polynomials in conjunction with a local linear stability analysis. We refer to these schemes as Chebyshev Parameterized Runge Kutta (CPRK) methods. CPRK methods of one to four stages are derived as functions of the parameters which describe an ellipse {Epsilon} which the stability domain of the methods is known to contain. Of particular interest are two-stage, first-order CPRK and four-stage, first-order methods. It is found that the former method can be identified with any two-stage RK method through the correct choice of parameters. The latter method is found to have a wide range of stability domains, with a maximum extension of 32 along the real axis. Recursion performance results are presented below for a model linear convection-diffusion problem as well as non-linear fluid flow problems discretized by both finite-difference and finite-element methods.

  16. Explicit formulation of a nodal transport method for discrete ordinates calculations in two-dimensional fixed-source problems

    Tres, Anderson [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Matematica Aplicada; Becker Picoloto, Camila [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Prolo Filho, Joao Francisco [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Inst de Matematica, Estatistica e Fisica; Dias da Cunha, Rudnei; Basso Barichello, Liliane [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Inst de Matematica


    In this work a study of two-dimensional fixed-source neutron transport problems, in Cartesian geometry, is reported. The approach reduces the complexity of the multidimensional problem using a combination of nodal schemes and the Analytical Discrete Ordinates Method (ADO). The unknown leakage terms on the boundaries that appear from the use of the derivation of the nodal scheme are incorporated to the problem source term, such as to couple the one-dimensional integrated solutions, made explicit in terms of the x and y spatial variables. The formulation leads to a considerable reduction of the order of the associated eigenvalue problems when combined with the usual symmetric quadratures, thereby providing solutions that have a higher degree of computational efficiency. Reflective-type boundary conditions are introduced to represent the domain on a simpler form than that previously considered in connection with the ADO method. Numerical results obtained with the technique are provided and compared to those present in the literature. (orig.)

  17. Explicit dynamics for numerical simulation of crack propagation by the extended finite element method; Dynamique explicite pour la simulation numerique de propagation de fissure par la methode des elements finis etendus

    Menouillard, T


    Computerized simulation is nowadays an integrating part of design and validation processes of mechanical structures. Simulation tools are more and more performing allowing a very acute description of the phenomena. Moreover, these tools are not limited to linear mechanics but are developed to describe more difficult behaviours as for instance structures damage which interests the safety domain. A dynamic or static load can thus lead to a damage, a crack and then a rupture of the structure. The fast dynamics allows to simulate 'fast' phenomena such as explosions, shocks and impacts on structure. The application domain is various. It concerns for instance the study of the lifetime and the accidents scenario of the nuclear reactor vessel. It is then very interesting, for fast dynamics codes, to be able to anticipate in a robust and stable way such phenomena: the assessment of damage in the structure and the simulation of crack propagation form an essential stake. The extended finite element method has the advantage to break away from mesh generation and from fields projection during the crack propagation. Effectively, crack is described kinematically by an appropriate strategy of enrichment of supplementary freedom degrees. Difficulties connecting the spatial discretization of this method with the temporal discretization of an explicit calculation scheme has then been revealed; these difficulties are the diagonal writing of the mass matrix and the associated stability time step. Here are presented two methods of mass matrix diagonalization based on the kinetic energy conservation, and studies of critical time steps for various enriched finite elements. The interest revealed here is that the time step is not more penalizing than those of the standard finite elements problem. Comparisons with numerical simulations on another code allow to validate the theoretical works. A crack propagation test in mixed mode has been exploited in order to verify the simulation

  18. Bound-Preserving Discontinuous Galerkin Methods for Conservative Phase Space Advection in Curvilinear Coordinates

    Endeve, Eirik; Xing, Yulong; Mezzacappa, Anthony


    We extend the positivity-preserving method of Zhang & Shu (2010, JCP, 229, 3091-3120) to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stability-preserving, Runge-Kutta (SSP-RK) time integration. Special care in taken to ensure that the method preserves strict bounds for the phase space distribution function $f$; i.e., $f\\in[0,1]$. The combination of suitable CFL conditions and the use of the high-order limiter proposed in Zhang & Shu (2010) is sufficient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergence-free property of ...

  19. The Cauchy-Lagrangian method for numerical analysis of Euler flow

    Podvigina, O; Frisch, U


    A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by simple recurrence relations that follow from the Cauchy invariants formulation of the Euler equations (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution. Tests performed on the two-dimensional Euler equation indicate that the Cauchy-Lagrangian method is more --- and occasionally much more --- efficient and less prone to instability than Eulerian Runge-Kutta methods and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algor...

  20. A first course in ordinary differential equations analytical and numerical methods

    Hermann, Martin


    This book presents a modern introduction to analytical and numerical techniques for solving ordinary differential equations (ODEs). Contrary to the traditional format—the theorem-and-proof format—the book is focusing on analytical and numerical methods. The book supplies a variety of problems and examples, ranging from the elementary to the advanced level, to introduce and study the mathematics of ODEs. The analytical part of the book deals with solution techniques for scalar first-order and second-order linear ODEs, and systems of linear ODEs—with a special focus on the Laplace transform, operator techniques and power series solutions. In the numerical part, theoretical and practical aspects of Runge-Kutta methods for solving initial-value problems and shooting methods for linear two-point boundary-value problems are considered. The book is intended as a primary text for courses on the theory of ODEs and numerical treatment of ODEs for advanced undergraduate and early graduate students. It is assumed t...

  1. Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method

    Souza de Paula, Aline [COPPE - Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, P.O. Box 68503, 21.941-972 Rio de Janeiro, RJ (Brazil)], E-mail:; Savi, Marcelo Amorim [COPPE - Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, P.O. Box 68503, 21.941-972 Rio de Janeiro, RJ (Brazil)], E-mail:


    Chaos control is employed for the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The extended time-delayed feedback control uses a continuous feedback loop incorporating information from previous states of the system in order to stabilize unstable orbits. This article deals with the chaos control of a nonlinear pendulum employing the extended time-delayed feedback control method. The control law leads to delay-differential equations (DDEs) that contain derivatives that depend on the solution of previous time instants. A fourth-order Runge-Kutta method with linear interpolation on the delayed variables is employed for numerical simulations of the DDEs and its initial function is estimated by a Taylor series expansion. During the learning stage, the UPOs are identified by the close-return method and control parameters are chosen for each desired UPO by defining situations where the largest Lyapunov exponent becomes negative. Analyses of a nonlinear pendulum are carried out by considering signals that are generated by numerical integration of the mathematical model using experimentally identified parameters. Results show the capability of the control procedure to stabilize UPOs of the dynamical system, highlighting some difficulties to achieve the stabilization of the desired orbit.

  2. Time domain finite element method for gyroscopic systems%陀螺系统时间有限元方法

    隋永枫; 高强; 钟万勰


    Based on the variational principle, a time domain finite element method for gyroscopic systems was presented. The corresponding trial function matrix, element stiffness matrix and inhomogeneous force were provided. The method inherits the property of symplectic conservation and enhances computational accuracy. An example comparing the numerical results obtained from three different methods: time domain FEM, 4th order Runge-Kutta method and Newmark method demonstrates the advantages of the time domain FEM.%将保辛的时间有限元方法应用于陀螺系统,扩展时间有限元方法的应用领域.同时导出陀螺系统时间有限元方法的形函数矩阵、时间单元刚度阵列式和非齐次外力的表达式.该方法既继承了有限元保辛的优良特性,又大大提高了数值计算精度,具有非常明显的优越性.算例给出本文方法、四阶Runge-Kutta方法与Newmark方法的比较结果,进一步表明本方法的优越性.

  3. Structure-preserving algorithms for the Duffing equation

    Gang Tie-Qiang; Mei Feng-Xiang; Xie Jia-Fang


    In this paper,the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε.Firstly,based on the gradient Hamiltonian decomposition theory of vector fields,by using splitting methods,this paper constructs structure-preserving algorithms(SPAs)for the Duffing equation.Then,according to the Liouville formula,it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation.However,considering the explicit Runge-Kutta methods,this paper finds that there is an error term of order p+1 for the Jacobian matrix determinants.The volume evolution law of a given region in phase space is discussed for different algorithms,respectively.As a result,the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper.Finally,through numerical experiments,relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method(a second-order Runge-Kutta method)are compared.Computational results illustrate that the SPAs are evidently better than the Heun method when ε is small or equal to zero.

  4. Finite water depth effect on wave-body problems solved by Rankine source method

    Feng, Aichun; Tang, Peng; You, Yunxiang; Liu, Kaizhou


    Finite water depth effect for wave-body problems are studied by continuous Rankine source method and non- desingularized technique. Free surface and seabed surface profiles are represented by continuous panels rather than a discretization by isolated points. These panels are positioned exactly on the fluid boundary surfaces and therefore no desingularization technique is required. Space increment method is applied for both free surface source and seabed source arrangements to reduce computational cost and improve numerical efficiency. Fourth order Runge-Kutta iteration scheme is adopted on the free surface updating at every time step. The finite water depth effect is studied quantitatively for a series of cylinders with different B/T ratios. The accuracy and efficiency of the proposed model are validated by comparison with published numerical results and experimental data. Numerical results show that hydrodynamic coefficients vary for cylinder bodies with different ratios of B/T. For certain set of B/T ratios the effect of finite water depth increases quickly with the increase of motion frequency and becomes stable when frequency is relatively large. It also shows that water depths have larger hydrodynamic effects on cylinder with larger breadth to draft ratios. Both the heave added mass and damping coefficients increase across the frequency range with the water depths decrease for forced heave motion. The water depths have smaller effects on sway motion response than on heave motion response.

  5. A method for the determination of the coefficient of rolling friction using cycloidal pendulum

    Ciornei, M. C.; Alaci, S.; Ciornei, F. C.; Romanu, I. C.


    The paper presents a method for experimental finding of coefficient of rolling friction appropriate for biomedical applications based on the theory of cycloidal pendulum. When a mobile circle rolls over a fixed straight line, the points from the circle describe trajectories called normal cycloids. To materialize this model, it is sufficient that a small region from boundary surfaces of a moving rigid body is spherical. Assuming pure rolling motion, the equation of motion of the cycloidal pendulum is obtained - an ordinary nonlinear differential equation. The experimental device is composed by two interconnected balls rolling over the material to be studied. The inertial characteristics of the pendulum can be adjusted via weights placed on a rod. A laser spot oscillates together to the pendulum and provides the amplitude of oscillations. After finding the experimental parameters necessary in differential equation of motion, it can be integrated using the Runge-Kutta of fourth order method. The equation was integrated for several materials and found values of rolling friction coefficients. Two main conclusions are drawn: the coefficient of rolling friction influenced significantly the amplitude of oscillation but the effect upon the period of oscillation is practically imperceptible. A methodology is proposed for finding the rolling friction coefficient and the pure rolling condition is verified.

  6. A high-order multi-zone cut-stencil method for numerical simulations of high-speed flows over complex geometries

    Greene, Patrick T.; Eldredge, Jeff D.; Zhong, Xiaolin; Kim, John


    In this paper, we present a method for performing uniformly high-order direct numerical simulations of high-speed flows over arbitrary geometries. The method was developed with the goal of simulating and studying the effects of complex isolated roughness elements on the stability of hypersonic boundary layers. The simulations are carried out on Cartesian grids with the geometries imposed by a third-order cut-stencil method. A fifth-order hybrid weighted essentially non-oscillatory scheme was implemented to capture any steep gradients in the flow created by the geometries and a third-order Runge-Kutta method is used for time advancement. A multi-zone refinement method was also utilized to provide extra resolution at locations with expected complex physics. The combination results in a globally fourth-order scheme in space and third order in time. Results confirming the method's high order of convergence are shown. Two-dimensional and three-dimensional test cases are presented and show good agreement with previous results. A simulation of Mach 3 flow over the logo of the Ubuntu Linux distribution is shown to demonstrate the method's capabilities for handling complex geometries. Results for Mach 6 wall-bounded flow over a three-dimensional cylindrical roughness element are also presented. The results demonstrate that the method is a promising tool for the study of hypersonic roughness-induced transition.

  7. Semi-computational simulation of magneto-hemodynamic flow in a semi-porous channel using optimal homotopy and differential transform methods.

    Basiri Parsa, A; Rashidi, M M; Anwar Bég, O; Sadri, S M


    In this paper, the semi-numerical techniques known as the optimal homotopy analysis method (HAM) and Differential Transform Method (DTM) are applied to study the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field. The two-dimensional momentum conservation partial differential equations are reduced to ordinary form incorporating Lorentizian magnetohydrodynamic body force terms. These ordinary differential equations are solved by the homotopy analysis method, the differential transform method and also a numerical method (fourth-order Runge-Kutta quadrature with a shooting method), under physically realistic boundary conditions. The homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of solution series. The differential transform method (DTM) does not require an auxiliary parameter and is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The influence of Hartmann number (Ha) and transpiration Reynolds number (mass transfer parameter, Re) on the velocity profiles in the channel are studied in detail. Interesting fluid dynamic characteristics are revealed and addressed. The HAM and DTM solutions are shown to both correlate well with numerical quadrature solutions, testifying to the accuracy of both HAM and DTM in nonlinear magneto-hemodynamics problems. Both these semi-numerical techniques hold excellent potential in modeling nonlinear viscous flows in biological systems.

  8. New matrix method for response analysis of circumferentially stiffened non-circular cylindrical shells under harmonic pressure


    Based on the governing equation of vibration of a kind of cylindrical shells written in a matrix differential equation of the first order, a new matrix method is presented for steady-state vibration analysis of a noncircular cylindrical shell simply supported at two ends and circumferentially stiffened by rings under harmonic pressure. Its difference from the existing works by Yamada and Irie is that the matrix differential equation is solved by using the extended homogeneous capacity precision integration approach other than the Runge-Kutta-Gill integration method. The transfer matrix can easily be determined by a high precision integration scheme. In addition, besides the normal interacting forces, which were commonly adopted by researchers earlier, the tangential interacting forces between the cylindrical shell and the rings are considered at the same time by means of the Dirac-δ function. The effects of the exciting frequencies on displacements and stresses responses have been investigated. Numerical results show that the proposed method is more efficient than the aforementioned method.

  9. A fully-coupled discontinuous Galerkin spectral element method for two-phase flow in petroleum reservoirs

    Taneja, Ankur; Higdon, Jonathan


    A spectral element method (SEM) is presented to simulate two-phase fluid flow (oil and water phase) in petroleum reservoirs. Petroleum reservoirs are porous media with heterogeneous geologic features, and the flow of two immiscible phases involves sharp, moving interfaces. The governing equations of motion are time-dependent, non-linear PDEs with strong hyperbolic nature. A fully-coupled numerical scheme using discontinuous Galerkin (DG) method with nodal spectral element basis functions for spatial discretization, and an implicit Runge-Kutta type time-stepping is developed to solve the PDEs in a robust, stable manner. Isoparameteric mapping is used to generate grids for reservoir and well geometry. We present the performance capabilities of the DG scheme with high-order basis functions to accurately resolve sharp fluid interfaces and a variety of heterogeneous geologic features. High-order convergence of SEM is demonstrated. Numerical results are presented for reservoir flows with various injection-production patterns. Typical reservoir heterogeneities like low-permeable regions, impermeable shale barriers, etc. are included in the numerical tests. Comparisons with commonly used finite volume methods and linear and quadratic finite element methods are presented. ExxonMobil Upstream Research Co.

  10. Radiative transfer equation for predicting light propagation in biological media: comparison of a modified finite volume method, the Monte Carlo technique, and an exact analytical solution.

    Asllanaj, Fatmir; Contassot-Vivier, Sylvain; Liemert, André; Kienle, Alwin


    We examine the accuracy of a modified finite volume method compared to analytical and Monte Carlo solutions for solving the radiative transfer equation. The model is used for predicting light propagation within a two-dimensional absorbing and highly forward-scattering medium such as biological tissue subjected to a collimated light beam. Numerical simulations for the spatially resolved reflectance and transmittance are presented considering refractive index mismatch with Fresnel reflection at the interface, homogeneous and two-layered media. Time-dependent as well as steady-state cases are considered. In the steady state, it is found that the modified finite volume method is in good agreement with the other two methods. The relative differences between the solutions are found to decrease with spatial mesh refinement applied for the modified finite volume method obtaining method is used for the time semi-discretization of the radiative transfer equation. An agreement among the modified finite volume method, Runge-Kutta method, and Monte Carlo solutions are shown, but with relative differences higher than in the steady state.

  11. On the Linear Stability of the Fifth-Order WENO Discretization

    Motamed, Mohammad


    We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge-Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge-Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis. © Springer Science+Business Media, LLC 2010.

  12. Z-Transform Techniques for Improved Real-Time Digital Simulation of Continuous Systems: Runge-Kutta Convolutions Adjusted for Unit Step Response via Pole-Residues.


    shortcut is available; note that on the right-hand side of Equation (26) the first term leads to Eular Convolution and the second to Mean Value...Convolution. Eular Convolution and Mean Value Convolution are just special cases of R-K(2,a) Convolution (see Table 2). TABLE 2. SPECIAL CASES OF R-K(2,a)C...Convolution Eular 0 Mean Value for 1/2 1/2 Trapezoidal I For a single real pole filter, F(s) - 1 (28) and any input, G(s), the approximation using R-K(2

  13. Adaptive Runge-Kutta integration for stiff systems: Comparing Nosé and Nosé-Hoover dynamics for the harmonic oscillator

    Graham Hoover, William; Clinton Sprott, Julien; Griswold Hoover, Carol


    We describe the application of adaptive (variable time step) integrators to stiff differential equations encountered in many applications. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely related examples, Nosé's dynamics and Nosé-Hoover dynamics, are both based on Hamiltonian mechanics and generate microstates consistent with Gibbs' canonical ensemble. Nosé's dynamics is stiff and can present severe numerical difficulties. Nosé-Hoover dynamics, although it follows exactly the same trajectory, is smooth and relatively trouble-free. We emphasize the power of adaptive integrators to resolve stiff problems such as the Nosé dynamics for the harmonic oscillator. The solutions also illustrate the power of computer graphics to enrich numerical solutions.

  14. Nonlinear Simulations of Coalescence Instability Using a Flux Difference Splitting Method

    Ma, Jun; Qin, Hong; Yu, Zhi; Li, Dehui


    A flux difference splitting numerical scheme based on the finite volume method is applied to study ideal/resistive magnetohydrodynamics. The ideal/resistive MHD equations are cast as a set of hyperbolic conservation laws, and we develop a numerical capability to solve the weak solutions of these hyperbolic conservation laws by combining a multi-state Harten-Lax-Van Leer approximate Riemann solver with the hyperbolic divergence cleaning technique, high order shock-capturing reconstruction schemes, and a third order total variance diminishing Runge-Kutta time evolving scheme. The developed simulation code is applied to study the long time nonlinear evolution of the coalescence instability. It is verified that small structures in the instability oscillate with time and then merge into medium structures in a coherent manner. The medium structures then evolve and merge into large structures, and this trend continues through all scale-lengths. The physics of this interesting nonlinear dynamics is numerically analyzed. supported by the National Magnetic Confinement Fusion Science Program of China (Nos. 2013GB111002, 2013GB105003, 2013GB111000, 2014GB124005, 2015GB111003), National Natural Science Foundation of China (Nos. 11305171, 11405208), JSPS-NRF-NSFC A3 Foresight Program in the field of Plasma Physics (NSFC-11261140328), the Science Foundation of the Institute of Plasma Physics, Chinese Academy of Sciences (DSJJ-15-JC02) and the CAS Program for the Interdisciplinary Collaboration Team

  15. Two-phase nanofluid condensation and heat transfer modeling using least square method (LSM) for industrial applications

    Hatami, M.; Mosayebidorcheh, S.; Jing, D.


    In this paper, two-phase Nanofluid condensation and heat transfer analysis over a vertical plate under gravity and between two parallel plates under magnetic force are investigated respectively using Least Square Method (LSM) and numerical method. After presenting the governing equations and solving them by LSM, the accuracy of results is examined by fourth order Runge-Kutta numerical method. Modeling results show that the condensate film thickness after condensation is reduced and therefore, the rate of heat transfer is enhanced by the addition of nanoparticles to the regular fluid. Effect of different nanoparticles and constant numbers on the temperature/velocity/concentration profiles as well as Nusselt number and boundary layer thickness, are also investigated. For instance, it was found that TiO2 and Ag have maximum boundary layer thicknesses and Nusselt number, respectively. By considering the magnetic field effect, it is also found that nanoparticles concentration can be controlled by changing the Hartmann number which, in turn, leads to different condensation and heat transfer properties.

  16. A Parallel 3D Spectral Difference Method for Solutions of Compressible Navier Stokes Equations on Deforming Grids and Simulations of Vortex Induced Vibration

    DeJong, Andrew

    Numerical models of fluid-structure interaction have grown in importance due to increasing interest in environmental energy harvesting, airfoil-gust interactions, and bio-inspired formation flying. Powered by increasingly powerful parallel computers, such models seek to explain the fundamental physics behind the complex, unsteady fluid-structure phenomena. To this end, a high-fidelity computational model based on the high-order spectral difference method on 3D unstructured, dynamic meshes has been developed. The spectral difference method constructs continuous solution fields within each element with a Riemann solver to compute the inviscid fluxes at the element interfaces and an averaging mechanism to compute the viscous fluxes. This method has shown promise in the past as a highly accurate, yet sufficiently fast method for solving unsteady viscous compressible flows. The solver is monolithically coupled to the equations of motion of an elastically mounted 3-degree of freedom rigid bluff body undergoing flow-induced lift, drag, and torque. The mesh is deformed using 4 methods: an analytic function, Laplace equation, biharmonic equation, and a bi-elliptic equation with variable diffusivity. This single system of equations -- fluid and structure -- is advanced through time using a 5-stage, 4th-order Runge-Kutta scheme. Message Passing Interface is used to run the coupled system in parallel on up to 240 processors. The solver is validated against previously published numerical and experimental data for an elastically mounted cylinder. The effect of adding an upstream body and inducing wake galloping is observed.

  17. Numerical simulation of magneto-acoustic Wave Phase Conjugation with the DG method in the CPR framework

    Modarreszadeh, Seyedamirreza; Timofeev, Evgeny; Merlen, Alain; Matar, Olivier Bou; Pernod, Philippe


    The present paper is concerned with the numerical modeling of magneto-acoustic Wave Phase Conjugation (WPC) phenomena. Since ultrasonic waves in the WPC applications have short wavelengths relative to the traveling distances, high-order numerical methods in both space and time domains are required. The numerical scheme chosen for the current research is the Runge-Kutta Discontinuous Galerkin (RKDG) method incorporated into the Correction Procedure via Reconstruction (CPR) framework. In order to avoid non-physical oscillations near high-gradient regions, a Weighted Essentially Non-Oscillatory (WENO) limiter is used to reconstruct the solutions in the affected cells. After being assured that the numerical scheme has appropriate accuracy and performance, the WPC process is modeled in both linear and non-linear regimes. The results in the linear regime are in acceptable agreement with the analytical solution. The only significant deviation between the linear and non-linear results is at the sensor within the passive zone, where the mean pressure starts to grow gradually in the non-linear regime due to overtaking of the low-velocity pressure waves by the high-velocity ones.

  18. Optimal locations of piezoelectric patches for supersonic flutter control of honeycomb sandwich panels, using the NSGA-II method

    Nezami, M.; Gholami, B.


    The active flutter control of supersonic sandwich panels with regular honeycomb interlayers under impact load excitation is studied using piezoelectric patches. A non-dominated sorting-based multi-objective evolutionary algorithm, called non-dominated sorting genetic algorithm II (NSGA-II) is suggested to find the optimal locations for different numbers of piezoelectric actuator/sensor pairs. Quasi-steady first order supersonic piston theory is employed to define aerodynamic loading and the p-method is applied to find the flutter bounds. Hamilton’s principle in conjunction with the generalized Fourier expansions and Galerkin method are used to develop the dynamical model of the structural systems in the state-space domain. The classical Runge-Kutta time integration algorithm is then used to calculate the open-loop aeroelastic response of the system. The maximum flutter velocity and minimum voltage applied to actuators are calculated according to the optimal locations of piezoelectric patches obtained using the NSGA-II and then the proportional feedback is used to actively suppress the closed loop system response. Finally the control effects, using the two different controllers, are compared.

  19. Development of polaron-transformed explicitly correlated full configuration interaction method for investigation of quantum-confined Stark effect in GaAs quantum dots

    Blanton, Christopher J; Chakraborty, Arindam


    The effect of external electric field on electron-hole correlation in GaAs quantum dots is investigated. The electron-hole Schrodinger equation in the presence of external electric field is solved using explicitly correlated full configuration interaction (XCFCI) method and accurate exciton binding energy and electron-hole recombination probability are obtained. The effect of the electric field was included in the 1-particle single component basis functions by performing variational polaron transformation. The quality of the wavefunction at small inter-particle distances was improved by using Gaussian-type geminal function that depended explicitly on the electron-hole separation distance. The parameters of the explicitly correlated function were determined variationally at each field strength. The scaling of total exciton energy, exciton binding energy, and electron-hole recombination probability with respect to the strength of the electric field was investigated. It was found that a 500 kV/cm change in elect...

  20. Dual Solutions for MHD Jeffery–Hamel Nano-Fluid Flow in Non-parallel Walls Using Predictor Homotopy Analysis Method

    Navid Freidoonimehr


    Full Text Available The main purpose of this study is to present dual solutions for the problem of magneto-hydrodynamic Jeffery–Hamel nano-fluid flow in non-parallel walls. To do so, we employ a new analytical technique, Predictor Homotopy Analysis Method (PHAM. This effective method is capable to calculate all branches of the multiple solutions simultaneously. Moreover, comparison of the PHAM results with numerical results obtained by the shooting method coupled with a Runge-Kutta integration method illustrates the high accuracy for this technique. For the current problem, it is found that the multiple (dual solutions exist for some values of governing parameters especially for the convergent channel cases (α = -1. The fluid in the non-parallel walls, divergent and convergent channels, is the drinking water containing different nanoparticles; Copper oxide (CuO, Copper (Cu and Silver (Ag. The effects of nanoparticle volume fraction parameter (φ, Reynolds number (Re, magnetic parameter (Mn, and angle of the channel (α as well as different types of nanoparticles on the flow characteristics are discussed.

  1. Nonlinear oscillation of nanoelectro-mechanical resonators using energy balance method: considering the size effect and the van der Waals force

    Ghalambaz, Mohammad; Ghalambaz, Mehdi; Edalatifar, Mohammad


    The energy balance method is utilized to analyze the oscillation of a nonlinear nanoelectro-mechanical system resonator. The resonator comprises an electrode, which is embedded between two substrates. Two types of clamped-clamped and cantilever nano-resonators are studied. The effects of the van der Waals attractions, Casimir force, the small size, the fringing field, the mid-plane stretching, and the axial load are taken into account. The governing partial differential equation of the resonator is reduced using the Galerkin method. The energy method is applied to obtain an analytical solution without considering any linearization or small parameter. The results of the present study are compared with the results available in the literature. In addition, the results of the present analytical solution are compared with the Runge-Kutta numerical results. An excellent agreement between the present analytical solution, numerical solution, and the results available in the literature was found. The influences of the van der Waals force, Casimir force, size effect, and fringing field effect on the oscillation frequency of resonators are studied. The results indicate that the presence of the intermolecular forces (van der Waals), Casimir force, and fringing field effect decreases the oscillation frequency of the resonator. In contrast, the presence of the size effect increases the oscillation frequency of the resonator.

  2. Faster Simulation Methods for the Nonstationary Random Vibrations of Non-linear MDOF Systems

    Askar, A.; Köylüo, U.; Nielsen, Søren R.K.


    . Such a treatment offers higher rates of convergence, faster speed and higher accuracy. These procedures are compared to the direct Monte Carlo simulation procedure, which uses a fourth order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process...

  3. Faster Simulation Methods for the Non-Stationary Random Vibrations of Non-Linear MDOF Systems

    Askar, A.; Köylüoglu, H. U.; Nielsen, Søren R. K.

    . Such a treatment offers higher rates of convergence, faster speed and higher accuracy. These procedures are compared to the direct Monte Carlo simulation procedure, which uses a fourth order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process...

  4. A Third-Order p-Laplacian Boundary Value Problem Solved by an SL(3,ℝ Lie-Group Shooting Method

    Chein-Shan Liu


    Full Text Available The boundary layer problem for power-law fluid can be recast to a third-order p-Laplacian boundary value problem (BVP. In this paper, we transform the third-order p-Laplacian into a new system which exhibits a Lie-symmetry SL(3,ℝ. Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of r∈[0,1]. The present SL(3,ℝ Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order p-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4 method to obtain a quite accurate numerical solution of the p-Laplacian.

  5. Investigation of acceleration effects on missile aerodynamics using computational fluid dynamics

    Gledhill, Irvy MA


    Full Text Available ) or upwind TVD flux difference splitting. An explicit Runge-Kutta local time-stepping is used for steady state calculations, and an implicit time-integration with dual time-stepping is used for the time accurate computations. To enhance the convergence... the slip airfoil surface in the dimensions modelled. A second order central difference scheme was used with Jameson dissipation [14], [13]. An implicit five stage Runge-Kutta scheme with backward Euler time differencing, 5 W-cycle multi-grid levels...

  6. Contribution to the asymptotic estimation of the global error of single step numerical integration methods. Application to the simulation of electric power networks; Contribution a l'estimation asymptotique de l'erreur globale des methodes d'integration numerique a un pas. Application a la simulation des reseaux electriques

    Aid, R.


    This work comes from an industrial problem of validating numerical solutions of ordinary differential equations modeling power systems. This problem is solved using asymptotic estimators of the global error. Four techniques are studied: Richardson estimator (RS), Zadunaisky's techniques (ZD), integration of the variational equation (EV), and Solving for the correction (SC). We give some precisions on the relative order of SC w.r.t. the order of the numerical method. A new variant of ZD is proposed that uses the Modified Equation. In the case of variable step-size, it is shown that under suitable restriction, on the hypothesis of the step-size selection, ZD and SC are still valid. Moreover, some Runge-Kutta methods are shown to need less hypothesis on the step-sizes to exhibit a valid order of convergence for ZD and SC. Numerical tests conclude this analysis. Industrial cases are given. Finally, an algorithm to avoid the a priori specification of the integration path for complex time differential equations is proposed. (author)

  7. 尾矿库调洪演算的数值方法及其应用研究%Numerical Method for Flood Regulating Calculus of Tailing Dam and Its Application

    曹金海; 段蔚平; 汪良峰


    将数值计算方法应用于尾矿库调洪演算中,采用四阶龙格-库塔求解尾矿库水量平衡微分方程与入泄关系,并采用插值法将库容曲线及排洪曲线解析化,实现了尾矿库调洪演算.应用fortran编程语言编制计算机程序进行实例计算,计算结果能达到预期目标.%The numerical calculation method was applied in flood regulating calculus of tailing dam. Fourth order Runge-Kutta was used to determine the relation between the differential equation of water balance of tailing dam and flood discharge. The interpolation method was adopted to analyze capacity curve and flood discharge curve to realize the flood regulating calculus of tailing dam. Programming language fortran was applied to program computer program and the calculation could reach the expected goal.

  8. Application of symplectic algorithms to QCT calculation:H + H2 system

    吴韬; 居宁; 沈长圣; 居冠之


    The basic theory of symplectic algorithm was introduced. A comparison between Runge-Kutta method and symplectic integration method was preformed in the simulation of the long time behavior of H + H2 system on BKMP potential energy surface. Our results reveal a dis-sipative behavior in the integral of ordinary differential equation by the fourth order Runge-Kutta method, which causes incorrect simulation results in QCT calculations. However, when the symplectic integration method is applied, the dissipative behavior is not found in the same system. When the initial state is the same, the energy deviation of fourth order symplectic integral method is almost one percent of that of fourth order Runge-Kutta method in a 60000-step simulation, and that of sixth order symplectic integral method is much less. These results show that the symplectic integral methods are always the better choice in the integral calculation of the long time behavior in maintaining energy conservation.

  9. Simulation methods with extended stability for stiff biochemical Kinetics

    Rué Pau


    Full Text Available Abstract Background With increasing computer power, simulating the dynamics of complex systems in chemistry and biology is becoming increasingly routine. The modelling of individual reactions in (biochemical systems involves a large number of random events that can be simulated by the stochastic simulation algorithm (SSA. The key quantity is the step size, or waiting time, τ, whose value inversely depends on the size of the propensities of the different channel reactions and which needs to be re-evaluated after every firing event. Such a discrete event simulation may be extremely expensive, in particular for stiff systems where τ can be very short due to the fast kinetics of some of the channel reactions. Several alternative methods have been put forward to increase the integration step size. The so-called τ-leap approach takes a larger step size by allowing all the reactions to fire, from a Poisson or Binomial distribution, within that step. Although the expected value for the different species in the reactive system is maintained with respect to more precise methods, the variance at steady state can suffer from large errors as τ grows. Results In this paper we extend Poisson τ-leap methods to a general class of Runge-Kutta (RK τ-leap methods. We show that with the proper selection of the coefficients, the variance of the extended τ-leap can be well-behaved, leading to significantly larger step sizes. Conclusions The benefit of adapting the extended method to the use of RK frameworks is clear in terms of speed of calculation, as the number of evaluations of the Poisson distribution is still one set per time step, as in the original τ-leap method. The approach paves the way to explore new multiscale methods to simulate (biochemical systems.

  10. Numerical analysis of the dynamic interaction between wheel set and turnout crossing using the explicit finite element method

    Xin, L.; Markine, V. L.; Shevtsov, I. Y.


    A three-dimensional (3-D) explicit dynamic finite element (FE) model is developed to simulate the impact of the wheel on the crossing nose. The model consists of a wheel set moving over the turnout crossing. Realistic wheel, wing rail and crossing geometries have been used in the model. Using this model the dynamic responses of the system such as the contact forces between the wheel and the crossing, crossing nose displacements and accelerations, stresses in rail material as well as in sleepers and ballast can be obtained. Detailed analysis of the wheel set and crossing interaction using the local contact stress state in the rail is possible as well, which provides a good basis for prediction of the long-term behaviour of the crossing (fatigue analysis). In order to tune and validate the FE model field measurements conducted on several turnouts in the railway network in the Netherlands are used here. The parametric study including variations of the crossing nose geometries performed here demonstrates the capabilities of the developed model. The results of the validation and parametric study are presented and discussed.

  11. A robust WENO scheme for nonlinear waves in a moving reference frame

    Kontos, Stavros; Bingham, Harry B.; Lindberg, Ole


    the linear WENO weights based on a Taylor series expansion is introduced. A simplified smoothness indicator is proposed and is shown to perform well. The scheme is combined with high-order explicit Runge-Kutta time integration and a dissipative Lax-Friedrichs-type flux to solve for nonlinear wave propagation...

  12. PLUTO code for computational Astrophysics: News and Developments

    Tzeferacos, P.; Mignone, A.


    We present an overview on recent developments and functionalities available with the PLUTO code for astrophysical fluid dynamics. The recent extension of the code to a conservative finite difference formulation and high order spatial discretization of the compressible equations of magneto-hydrodynamics (MHD), complementary to its finite volume approach, allows for a highly accurate treatment of smooth flows, while avoiding loss of accuracy near smooth extrema and providing sharp non-oscillatory transitions at discontinuities. Among the novel features, we present alternative, fully explicit treatments to include non-ideal dissipative processes (namely viscosity, resistivity and anisotropic thermal conduction), that do not suffer from the usual timestep limitation of explicit time stepping. These methods, offsprings of the multistep Runge-Kutta family that use a Chebyshev polynomial recursion, are competitive substitutes of computationally expensive implicit schemes that involve sparse matrix inversion. Several multi-dimensional benchmarks and appli-cations assess the potential of PLUTO to efficiently handle many astrophysical problems.

  13. TIGER2 with solvent energy averaging (TIGER2A): An accelerated sampling method for large molecular systems with explicit representation of solvent.

    Li, Xianfeng; Snyder, James A; Stuart, Steven J; Latour, Robert A


    The recently developed "temperature intervals with global exchange of replicas" (TIGER2) accelerated sampling method is found to have inaccuracies when applied to systems with explicit solvation. This inaccuracy is due to the energy fluctuations of the solvent, which cause the sampling method to be less sensitive to the energy fluctuations of the solute. In the present work, the problem of the TIGER2 method is addressed in detail and a modification to the sampling method is introduced to correct this problem. The modified method is called "TIGER2 with solvent energy averaging," or TIGER2A. This new method overcomes the sampling problem with the TIGER2 algorithm and is able to closely approximate Boltzmann-weighted sampling of molecular systems with explicit solvation. The difference in performance between the TIGER2 and TIGER2A methods is demonstrated by comparing them against analytical results for simple one-dimensional models, against replica exchange molecular dynamics (REMD) simulations for sampling the conformation of alanine dipeptide and the folding behavior of (AAQAA)3 peptide in aqueous solution, and by comparing their performance in sampling the behavior of hen egg-white lysozyme in aqueous solution. The new TIGER2A method solves the problem caused by solvent energy fluctuations in TIGER2 while maintaining the two important characteristics of TIGER2, i.e., (1) using multiple replicas sampled at different temperature levels to help systems efficiently escape from local potential energy minima and (2) enabling the number of replicas used for a simulation to be independent of the size of the molecular system, thus providing an accelerated sampling method that can be used to efficiently sample systems considered too large for the application of conventional temperature REMD.

  14. Comparing Numerical Integration Schemes for Time-Continuous Car-Following Models

    Treiber, Martin


    When simulating trajectories by integrating time-continuous car-following models, standard integration schemes such as the forth-order Runge-Kutta method (RK4) are rarely used while the simple Euler's method is popular among researchers. We compare four explicit methods: Euler's method, ballistic update, Heun's method (trapezoidal rule), and the standard forth-order RK4. As performance metrics, we plot the global discretization error as a function of the numerical complexity. We tested the methods on several time-continuous car-following models in several multi-vehicle simulation scenarios with and without discontinuities such as stops or a discontinuous behavior of an external leader. We find that the theoretical advantage of RK4 (consistency order~4) only plays a role if both the acceleration function of the model and the external data of the simulation scenario are sufficiently often differentiable. Otherwise, we obtain lower (and often fractional) consistency orders. Although, to our knowledge, Heun's met...

  15. 2–stage stochastic Runge–Kutta for stochastic delay differential equations

    Rosli, Norhayati; Jusoh Awang, Rahimah [Faculty of Industrial Science and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300, Gambang, Pahang (Malaysia); Bahar, Arifah; Yeak, S. H. [Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia)


    This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.

  16. High-order Hamiltonian splitting for Vlasov-Poisson equations

    Casas, Fernando; Faou, Erwan; Mehrenberger, Michel


    We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions, we study the order conditions of such methods. It appears that these conditions are of Runge-Kutta-Nystr{\\"o}m type. In the one dimensional case, the order conditions can be further simplified, and efficient methods of order 6 with a reduced number of stages can be constructed. In the general case, high-order methods can also be constructed using explicit computations of commutators. Numerical results are performed and show the benefit of using high-order splitting schemes in that context. Complete and self-contained proofs of convergence results and rigorous error estimates are also given.

  17. Flexible CDOCKER: Development and application of a pseudo-explicit structure-based docking method within CHARMM.

    Gagnon, Jessica K; Law, Sean M; Brooks, Charles L


    Protein-ligand docking is a commonly used method for lead identification and refinement. While traditional structure-based docking methods represent the receptor as a rigid body, recent developments have been moving toward the inclusion of protein flexibility. Proteins exist in an interconverting ensemble of conformational states, but effectively and efficiently searching the conformational space available to both the receptor and ligand remains a well-appreciated computational challenge. To this end, we have developed the Flexible CDOCKER method as an extension of the family of complete docking solutions available within CHARMM. This method integrates atomically detailed side chain flexibility with grid-based docking methods, maintaining efficiency while allowing the protein and ligand configurations to explore their conformational space simultaneously. This is in contrast to existing approaches that use induced-fit like sampling, such as Glide or Autodock, where the protein or the ligand space is sampled independently in an iterative fashion. Presented here are developments to the CHARMM docking methodology to incorporate receptor flexibility and improvements to the sampling protocol as demonstrated with re-docking trials on a subset of the CCDC/Astex set. These developments within CDOCKER achieve docking accuracy competitive with or exceeding the performance of other widely utilized docking programs.

  18. A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and parallel

    Ketcheson, David I.


    We compare the three main types of high-order one-step initial value solvers: extrapolation, spectral deferred correction, and embedded Runge–Kutta pairs. We consider orders four through twelve, including both serial and parallel implementations. We cast extrapolation and deferred correction methods as fixed-order Runge–Kutta methods, providing a natural framework for the comparison. The stability and accuracy properties of the methods are analyzed by theoretical measures, and these are compared with the results of numerical tests. In serial, the eighth-order pair of Prince and Dormand (DOP8) is most efficient. But other high-order methods can be more efficient than DOP8 when implemented in parallel. This is demonstrated by comparing a parallelized version of the wellknown ODEX code with the (serial) DOP853 code. For an N-body problem with N = 400, the experimental extrapolation code is as fast as the tuned Runge–Kutta pair at loose tolerances, and is up to two times as fast at tight tolerances.

  19. A Theoretical Method for Characterizing Nonlinear Effects in Paul Traps with Added Octopole Field.

    Xiong, Caiqiao; Zhou, Xiaoyu; Zhang, Ning; Zhan, Lingpeng; Chen, Yongtai; Chen, Suming; Nie, Zongxiu


    In comparison with numerical methods, theoretical characterizations of ion motion in the nonlinear Paul traps always suffer from low accuracy and little applicability. To overcome the difficulties, the theoretical harmonic balance (HB) method was developed, and was validated by the numerical fourth-order Runge-Kutta (4th RK) method. Using the HB method, analytical ion trajectory and ion motion frequency in the superimposed octopole field, ε, were obtained by solving the nonlinear Mathieu equation (NME). The obtained accuracy of the HB method was comparable with that of the 4th RK method at the Mathieu parameter, q = 0.6, and the applicable q values could be extended to the entire first stability region with satisfactory accuracy. Two sorts of nonlinear effects of ion motion were studied, including ion frequency shift, Δβ, and ion amplitude variation, Δ(C(2n)/C0) (n ≠ 0). New phenomena regarding Δβ were observed, although extensive studies have been performed based on the pseudo-potential well (PW) model. For instance, the |Δβ| at ε = 0.1 and ε = -0.1 were found to be different, but they were the same in the PW model. This is the first time the nonlinear effects regarding Δ(C(2n)/C0) (n ≠ 0) are studied, and the associated study has been a challenge for both theoretical and numerical methods. The nonlinear effects of Δ(C(2n)/C0) (n ≠ 0) and Δβ were found to share some similarities at q < 0.6: both of them were proportional to ε, and the square of the initial ion displacement, z(0)(2).

  20. Numerical stability of finite difference algorithms for electrochemical kinetic simulations: Matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods and typical problems involving mixed boundary conditions

    Bieniasz, Leslaw K.; Østerby, Ole; Britz, Dieter


    The stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference discretizations of example diffusional initial boundary value problems from electrochemical kinetics has been investigated using the matrix method of stability analysis. Special attention...

  1. SparseMaps—A systematic infrastructure for reduced-scaling electronic structure methods. IV. Linear-scaling second-order explicitly correlated energy with pair natural orbitals

    Pavošević, Fabijan; Pinski, Peter; Riplinger, Christoph; Neese, Frank; Valeev, Edward F.


    We present a formulation of the explicitly correlated second-order Møller-Plesset (MP2-F12) energy in which all nontrivial post-mean-field steps are formulated with linear computational complexity in system size. The two key ideas are the use of pair-natural orbitals for compact representation of wave function amplitudes and the use of domain approximation to impose the block sparsity. This development utilizes the concepts for sparse representation of tensors described in the context of the domain based local pair-natural orbital-MP2 (DLPNO-MP2) method by us recently [Pinski et al., J. Chem. Phys. 143, 034108 (2015)]. Novel developments reported here include the use of domains not only for the projected atomic orbitals, but also for the complementary auxiliary basis set (CABS) used to approximate the three- and four-electron integrals of the F12 theory, and a simplification of the standard B intermediate of the F12 theory that avoids computation of four-index two-electron integrals that involve two CABS indices. For quasi-1-dimensional systems (n-alkanes), the O (" separators="N ) DLPNO-MP2-F12 method becomes less expensive than the conventional O (" separators="N5 ) MP2-F12 for n between 10 and 15, for double- and triple-zeta basis sets; for the largest alkane, C200H402, in def2-TZVP basis, the observed computational complexity is N˜1.6, largely due to the cubic cost of computing the mean-field operators. The method reproduces the canonical MP2-F12 energy with high precision: 99.9% of the canonical correlation energy is recovered with the default truncation parameters. Although its cost is significantly higher than that of DLPNO-MP2 method, the cost increase is compensated by the great reduction of the basis set error due to explicit correlation.

  2. Sampling conformational space of intrinsically disordered proteins in explicit solvent: Comparison between well-tempered ensemble approach and solute tempering method.

    Han, Mengzhi; Xu, Ji; Ren, Ying


    Intrinsically disordered proteins (IDPs) are a class of proteins that expected to be largely unstructured under physiological conditions. Due to their heterogeneous nature, experimental characterization of IDP is challenging. Temperature replica exchange molecular dynamics (T-REMD) is a widely used enhanced sampling method to probe structural characteristics of these proteins. However, its application has been hindered due to its tremendous computational cost, especially when simulating large systems in explicit solvent. Two methods, parallel tempering well-tempered ensemble (PT-WTE) and replica exchange with solute tempering (REST), have been proposed to alleviate the computational expense of T-REMD. In this work, we select three different IDP systems to compare the sampling characteristics and efficiencies of the two methods Both the two methods could efficiently sample the conformational space of IDP and yield highly consistent results for all the three IDPs. The efficiencies of the two methods: are compatible, with about 5-6 times better than the plain T-REMD. Besides, the advantages and disadvantages of each method are also discussed. Specially, the PT-WTE method could provide temperature dependent data of the system which could not be achieved by REST, while the REST method could readily be used to a part of the system, which is quite efficient to simulate some biological processes. Copyright © 2016 Elsevier Inc. All rights reserved.

  3. Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

    Bich, Dao Huy; Xuan Nguyen, Nguyen


    In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge-Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency-amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.

  4. Reflection-free finite volume Maxwell's solver for adaptive grids

    Elkina, Nina


    We present a non-staggered method for the Maxwell equations in adaptively refined grids. The code is based on finite volume central scheme that preserves in a discrete form both divergence-free property of magnetic field and the Gauss law. High spatial accuracy is achieved with help of non-oscillatory extrema preserving piece-wise or piece-wise-quadratic reconstructions. The semi-discrete equations are solved by implicit-explicit Runge-Kutta method. The new adaptive grid Maxwell's solver is examined based on several 1d examples, including the an propagation of a Gaussian pulse through vacuum and partially ionised gas. Two-dimensional extension is tested with a Gaussian pulse incident on dielectric disc. Additionally, we focus on testing computational accuracy and efficiency.

  5. On the nonlinear dynamics of the traveling-wave solutions of the Serre equations

    Mitsotakis, Dimitrios; Carter, John D


    In this paper, we study numerically nonlinear phenomena related to the dynamics of the traveling wave solutions of the Serre equations including their stability, their persistence, resolution into solitary waves, and wave breaking. Other forms of solutions such as DSWs, are also considered. Some differences between the solutions of the Serre equations and the full Euler equations are also studied. Euler solitary waves propagate without large variations in shape when they are used as initial conditions in the Serre equations. The nonlinearities seem to play a crucial role in the generation of small-amplitude waves and appear to cause a recurrence phenomenon in linearly unstable solutions. The numerical method used in the paper utilizes a high order FEM with smooth, periodic splines in space and explicit Runge-Kutta methods in time. The solutions of the Serre system are compared with the corresponding ones of the asymptotically-related Euler system whenever is possible.

  6. Nodal DG-FEM solution of high-order Boussinesq-type equations

    Engsig-Karup, Allan Peter; Hesthaven, Jan S.; Bingham, Harry B.;


    We present a discontinuous Galerkin finite element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one and two horizontal dimensions. The continuous equations are discretized using nodal polynomial basis...... functions of arbitrary order in space on each element of an unstructured computational domain. A fourth order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy...... and convergence of the model with both h (grid size) and p (order) refinement are verified for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar; and reflection of a steep solitary wave from a vertical wall...

  7. Second-Order Accurate Projective Integrators for Multiscale Problems

    Lee, S L; Gear, C W


    We introduce new projective versions of second-order accurate Runge-Kutta and Adams-Bashforth methods, and demonstrate their use as outer integrators in solving stiff differential systems. An important outcome is that the new outer integrators, when combined with an inner telescopic projective integrator, can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators. If the stiff differential equations are not directly available, our formulations and stability analysis are general enough to allow the combined outer-inner projective integrators to be applied to black-box legacy codes or perform a coarse-grained time integration of microscopic systems to evolve macroscopic behavior, for example.

  8. Computation of three-dimensional, rotational flow through turbomachinery blade rows for improved aerodynamic design studies

    Subramanian, S. V.; Bozzola, R.; Povinelli, L. A.


    The performance of a three dimensional computer code developed for predicting the flowfield in stationary and rotating turbomachinery blade rows is described in this study. The four stage Runge-Kutta numerical integration scheme is used for solving the governing flow equations and yields solution to the full, three dimensional, unsteady Euler equations in cylindrical coordinates. This method is fully explicit and uses the finite volume, time marching procedure. In order to demonstrate the accuracy and efficiency of the code, steady solutions were obtained for several cascade geometries under widely varying flow conditions. Computed flowfield results are presented for a fully subsonic turbine stator and a low aspect ratio, transonic compressor rotor blade under maximum flow and peak efficiency design conditions. Comparisons with Laser Anemometer measurements and other numerical predictions are also provided to illustrate that the present method predicts important flow features with good accuracy and can be used for cost effective aerodynamic design studies.

  9. The development of three-dimensional adjoint method for flow control with blowing in convergent-divergent nozzle flows

    Sikarwar, Nidhi

    multiple experiments or numerical simulations. Alternatively an inverse design method can be used. An adjoint optimization method can be used to achieve the optimum blowing rate. It is shown that the method works for both geometry optimization and active control of the flow in order to deflect the flow in desirable ways. An adjoint optimization method is described. It is used to determine the blowing distribution in the diverging section of a convergent-divergent nozzle that gives a desired pressure distribution in the nozzle. Both the direct and adjoint problems and their associated boundary conditions are developed. The adjoint method is used to determine the blowing distribution required to minimize the shock strength in the nozzle to achieve a known target pressure and to achieve close to an ideally expanded flow pressure. A multi-block structured solver is developed to calculate the flow solution and associated adjoint variables. Two and three-dimensional calculations are performed for internal and external of the nozzle domains. A two step MacCormack scheme based on predictor- corrector technique is was used for some calculations. The four and five stage Runge-Kutta schemes are also used to artificially march in time. A modified Runge-Kutta scheme is used to accelerate the convergence to a steady state. Second order artificial dissipation has been added to stabilize the calculations. The steepest decent method has been used for the optimization of the blowing velocity after the gradients of the cost function with respect to the blowing velocity are calculated using adjoint method. Several examples are given of the optimization of blowing using the adjoint method.

  10. Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research.

    Wu, Hulin; Xue, Hongqi; Kumar, Arun


    Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches.

  11. A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes

    Zhang, Wensheng; Zhuang, Yuan; Zhang, Lina


    In this paper, we proposed a new efficient high-order finite volume method for 3D elastic wave simulation on unstructured tetrahedral meshes. With the relative coarse tetrahedral meshes, we make subdivision in each tetrahedron to generate a stencil for the high-order polynomial reconstruction. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. We perform the reconstruction on this stencil by using cell-averaged quantities based on the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, our new method has a very local property like DG and can be written as an inner-split computational scheme which is beneficial to reducing computational amount. Moreover, the stencil in our method is easy to generate for all tetrahedrons especially in the three-dimensional case. The resulting reconstruction matrix is invertible and remains unchanged for all tetrahedrons and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. We show convergence results obtained with the proposed method up to fifth order accuracy in space. The high-order accuracy in time is obtained by the Runge-Kutta method. Comparisons between numerical and analytic solutions show the proposed method can provide accurate wavefield information. Numerical simulation for a realistic model with complex topography demonstrates the effectiveness and potential applications of our method. Though the method is proposed based on the 3D elastic wave equation, it can be extended to other linear hyperbolic system.

  12. General-Relativistic Resistive Magnetohydrodynamics in three dimensions: formulation and tests

    Dionysopoulou, Kyriaki; Palenzuela, Carlos; Rezzolla, Luciano; Giacomazzo, Bruno


    We present a new numerical implementation of the general-relativistic resistive magnetohydrodynamics (MHD) equations within the Whisky code. The numerical method adopted exploits the properties of Implicit-Explicit Runge-Kutta numerical schemes to treat the stiff terms that appear in the equations for small electrical conductivities. Using tests in one, two, and three dimensions, we show that our implementation is robust and recovers the ideal-MHD limit in regimes of very high conductivity. Moreover, the results illustrate that the code is capable of describing physical setups in all ranges of conductivities. In addition to tests in flat spacetime, we report simulations of magnetized nonrotating relativistic stars, both in the Cowling approximation and in dynamical spacetimes. Finally, because of its astrophysical relevance and because it provides a severe testbed for general-relativistic codes with dynamical electromagnetic fields, we study the collapse of a nonrotating star to a black hole. We show that als...

  13. Toward a Consistent Framework for High Order Mesh Refinement Schemes in Numerical Relativity

    Mongwane, Bishop


    It has now become customary in the field of numerical relativity to couple high order finite difference schemes to mesh refinement algorithms. To this end, different modifications to the standard Berger-Oliger adaptive mesh refinement algorithm have been proposed. In this work we present a fourth order stable mesh refinement scheme with sub-cycling in time for numerical relativity. We do not use buffer zones to deal with refinement boundaries but explicitly specify boundary data for refined grids. We argue that the incompatibility of the standard mesh refinement algorithm with higher order Runge Kutta methods is a manifestation of order reduction phenomena, caused by inconsistent application of boundary data in the refined grids. Our scheme also addresses the problem of spurious reflections that are generated when propagating waves cross mesh refinement boundaries. We introduce a transition zone on refined levels within which the phase velocity of propagating modes is allowed to decelerate in order to smoothl...

  14. Gpu Implementation of a Viscous Flow Solver on Unstructured Grids

    Xu, Tianhao; Chen, Long


    Graphics processing units have gained popularities in scientific computing over past several years due to their outstanding parallel computing capability. Computational fluid dynamics applications involve large amounts of calculations, therefore a latest GPU card is preferable of which the peak computing performance and memory bandwidth are much better than a contemporary high-end CPU. We herein focus on the detailed implementation of our GPU targeting Reynolds-averaged Navier-Stokes equations solver based on finite-volume method. The solver employs a vertex-centered scheme on unstructured grids for the sake of being capable of handling complex topologies. Multiple optimizations are carried out to improve the memory accessing performance and kernel utilization. Both steady and unsteady flow simulation cases are carried out using explicit Runge-Kutta scheme. The solver with GPU acceleration in this paper is demonstrated to have competitive advantages over the CPU targeting one.

  15. Multidimensional Hall magnetohydrodynamics with isotropic or anisotropic thermal pressure: numerical scheme and its validation using solitary waves

    Strumik, Marek


    We present a numerical solver for plasma dynamics simulations in Hall magnetohydrodynamic (HMHD) approximation in one, two and three dimensions. We consider both isotropic and anisotropic thermal pressure cases, where a general gyrotropic approximation is used. Both explicit energy conservation equation and general polytropic state equations are considered. The numerical scheme incorporates second-order Runge-Kutta advancing in time and Kurganov-Tadmor scheme with van Leer flux limiter for the approximation of fluxes. A flux-interpolated constrained-transport approach is used to preserve solenoidal magnetic field in the simulations. The implemented code is validated using several test problems previously described in the literature. Additionally, we propose a new validation method for HMHD codes based on solitary waves that provides a possibility of quantitative rigorous testing in nonlinear (large amplitude) regime as an extension to standard tests using small-amplitude whistler waves. Quantitative tests of ...

  16. Entropy Generation Analysis of Open Parallel Microchannels Embedded Within a Permeable Continuous Moving Surface: Application to Magnetohydrodynamics (MHD

    Mohammad H. Yazdi


    Full Text Available This paper presents a new design of open parallel microchannels embedded within a permeable continuous moving surface due to reduction of exergy losses in magnetohydrodynamic (MHD flow at a prescribed surface temperature (PST. The entropy generation number is formulated by an integral of the local rate of entropy generation along the width of the surface based on an equal number of microchannels and no-slip gaps interspersed between those microchannels. The velocity, the temperature, the velocity gradient and the temperature gradient adjacent to the wall are substituted into this equation resulting from the momentum and energy equations obtained numerically by an explicit Runge-Kutta (4, 5 formula, the Dormand-Prince pair and shooting method. The entropy generation number, as well as the Bejan number, for various values of the involved parameters of the problem are also presented and discussed in detail.

  17. Study of Particle Rotation Effect in Gas-Solid Flows using Direct Numerical Simulation with a Lattice Boltzmann Method

    Kwon, Kyung [Tuskegee Univ., Tuskegee, AL (United States); Fan, Liang-Shih [The Ohio State Univ., Columbus, OH (United States); Zhou, Qiang [The Ohio State Univ., Columbus, OH (United States); Yang, Hui [The Ohio State Univ., Columbus, OH (United States)


    A new and efficient direct numerical method with second-order convergence accuracy was developed for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method combines the state-of-the-art immersed boundary method (IBM), the multi-direct forcing method, and the lattice Boltzmann method (LBM). First, the multi-direct forcing method is adopted in the improved IBM to better approximate the no-slip/no-penetration (ns/np) condition on the surface of particles. Second, a slight retraction of the Lagrangian grid from the surface towards the interior of particles with a fraction of the Eulerian grid spacing helps increase the convergence accuracy of the method. An over-relaxation technique in the procedure of multi-direct forcing method and the classical fourth order Runge-Kutta scheme in the coupled fluid-particle interaction were applied. The use of the classical fourth order Runge-Kutta scheme helps the overall IB-LBM achieve the second order accuracy and provides more accurate predictions of the translational and rotational motion of particles. The preexistent code with the first-order convergence rate is updated so that the updated new code can resolve the translational and rotational motion of particles with the second-order convergence rate. The updated code has been validated with several benchmark applications. The efficiency of IBM and thus the efficiency of IB-LBM were improved by reducing the number of the Lagragian markers on particles by using a new formula for the number of Lagrangian markers on particle surfaces. The immersed boundary-lattice Boltzmann method (IBLBM) has been shown to predict correctly the angular velocity of a particle. Prior to examining drag force exerted on a cluster of particles, the updated IB-LBM code along with the new formula for the number of Lagrangian markers has been further validated by solving several theoretical problems. Moreover, the unsteadiness of the drag force is examined when a

  18. Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

    Woollands, Robyn M.; Read, Julie L.; Probe, Austin B.; Junkins, John L.


    We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified Chebyshev-Picard iteration. The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not required, and instead it makes use of a reference trajectory and a set of n particular solutions. Any numerical integrator can be used for solving two-point boundary problems with the method of particular solutions, however we show that using modified Chebyshev-Picard iteration affords an avenue for increased efficiency that is not available with other step-by-step integrators. We take advantage of the path approximation nature of modified Chebyshev-Picard iteration (nodes iteratively converge to fixed points in space) and utilize a variable fidelity force model for propagating the reference trajectory. Remarkably, we demonstrate that computing the particular solutions with only low fidelity function evaluations greatly increases the efficiency of the algorithm while maintaining machine precision accuracy. Our study reveals that solving the perturbed Lambert's problem using the method of particular solutions with modified Chebyshev-Picard iteration is about an order of magnitude faster compared with the classical shooting method and a tenth-twelfth order Runge-Kutta integrator. It is well known that the solution to Lambert's problem over multiple revolutions is not unique and to ensure that all possible solutions are considered we make use of a reliable preexisting Keplerian Lambert solver to warm start our perturbed algorithm.

  19. Linear scaling explicitly correlated MP2-F12 and ONIOM methods for the long-range interactions of the nanoscale clusters in methanol aqueous solutions.

    Li, Wei


    A linear scaling quantum chemistry method, generalized energy-based fragmentation (GEBF) approach has been extended to the explicitly correlated second-order Møller-Plesset perturbation theory F12 (MP2-F12) method and own N-layer integrated molecular orbital molecular mechanics (ONIOM) method, in which GEBF-MP2-F12, GEBF-MP2, and conventional density functional tight-binding methods could be used for different layers. Then the long-range interactions in dilute methanol aqueous solutions are studied by computing the binding energies between methanol molecule and water molecules in gas-phase and condensed phase methanol-water clusters with various sizes, which were taken from classic molecular dynamics (MD) snapshots. By comparing with the results of force field methods, including SPC, TIP3P, PCFF, and AMOEBA09, the GEBF-MP2-F12 and GEBF-ONIOM methods are shown to be powerful and efficient for studying the long-range interactions at a high level. With the GEBF-ONIOM(MP2-F12:MP2) and GEBF-ONIOM(MP2-F12:MP2:cDFTB) methods, the diameters of the largest nanoscale clusters under studies are about 2.4 nm (747 atoms and 10 209 basis functions with aug-cc-pVDZ basis set) and 4 nm (3351 atoms), respectively, which are almost impossible to be treated by conventional MP2 or MP2-F12 method. Thus, the GEBF-F12 and GEBF-ONIOM methods are expected to be a practical tool for studying the nanoscale clusters in condensed phase, providing an alternative benchmark for ab initio and density functional theory studies, and developing new force fields by combining with classic MD simulations.

  20. 滤波策略对噪声场影响的研究%Study on the Impact of Filtering Strategies for Noise Calculation

    涂运冲; 甘加业; 吴克启


    In this paper, the impact of different filtering strategies for noise calculation is in- vestigated through direct numerical simulation of two-dimensional axisymmetric flow field. It is based on fourth order Runge-Kutta schemes for temporal discretization and sixth order compact finite-difference schemes for spatial discretization coupled with up to lOth-order low-pass filters which is adopted to control numerical stability. The suppression of the parasitic waves solutions were evaluated through comparison between explicit 10th-order and compact 10th-order filters,low-storage third-order Runge-Kutta scheme with four-step fourth-order Runge-Kutta scheme, the Parity grid method and the staggered grid method. The results show that in many above scenarios of inhibi- tion of parasitic waves, the calculation process of using staggered grid is most stable and most effective.%本文通过对二维轴对称射流场进行直接数值模拟,探讨了不同滤波策略对噪声场的影响。空间导数的离散采用六阶紧致差分格式,数值非稳定性的控制采用紧致10阶低滤波格式,时间推进采用四步四阶Runge—Kutta格式。对比分析了显式10阶与紧致10阶滤波格式,低存储3阶Runge—Kutta格式与四步四阶Runge—Kutta格式,交错网格与同位网格方法等几种抑制寄生波方案。分析结果表明,在上述诸多抑制寄生波的方案中,采用交错网格法,计算过程最稳定、最有效。

  1. Two-dimensional finite volume method for dam-break flow simulation



    A numerical model based upon a second-order upwind cell-center finite volume method on unstructured triangular grids is developed for solving shallow water equations.The assumption of a small depth downstream instead of a dry bed situation changes the wave structure and the propagation speed of the front which leads to incorrect results.The use of Harten-Lax-vau Leer (HLL) allows handling of wet/dry treatment.By usage of the HLL approximate Riemann solver,also it make possible to handle discontinuous solutions.As the assumption of a very small depth downstream of the dam can change the nature of the dam break flow problem which leads to incorrect results,the HLL approximate Riemann solver is used for the computation of inviscid flux functions,which makes it possible to handle discontinuous solutions.A multidimensional slope-limiting technique is applied to achieve second-order spatial accuracy and to prevent spurious oscillations.To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary,the friction source terms are treated in a fully implicit way.A third-order Runge-Kutta method is used for the time integration of semi-discrete equations.The developed numerical model has been applied to several test cases as well as to real flows.The tests are tested in two cases:oblique hydraulic jump and experimental dam break in converging-diverging flume.Numerical tests proved the robustness and accuracy of the model.The model has been applied for simulation of dam break analysis of Torogh in Irun.And finally the results have been used in preparing EAP (Emergency Action Plan).

  2. Explicit Content Image Detection

    Jorge Alberto Marcial Basilio


    Full Text Available This paper proposes a system gives for explicit content image detection based on Computer VisionAlgorithms, pattern recognition and FTK software Explicit Image Detection. In the first stage, HSV colormodel is used for the input images for the purpose of discriminating elements that are not human skinimages. Then the image is filtered using skin detection. The output image only contains the areas of whichit is composed. The results show a comparison between the proposed system and the company softwareAccess Data called Forensic Toolkit 3.1 Explicit Image Detection isperformed.

  3. Explicit Content Image Detection

    Jorge Alberto Marcial Basilio


    Full Text Available This paper proposes a system gives for explicit content image detection based on Computer Vision Algorithms, pattern recognition and FTK software Explicit Image Detection. In the first stage, HSV color model is used for the input images for the purpose of discriminating elements that are not human skin images. Then the image is filtered using skin detection. The output image only contains the areas of which it is composed. The results show a comparison between the proposed system and the company software Access Data called Forensic Toolkit 3.1 Explicit Image Detection isperformed.

  4. An adjoint method for a high-order discretization of deforming domain conservation laws for optimization of flow problems

    Zahr, M. J.; Persson, P.-O.


    The fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high-order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain is transformed into one on a fixed reference domain by the introduction of a time-dependent mapping that encapsulates the domain deformation and parametrization, resulting in an Arbitrary Lagrangian-Eulerian form of the governing equations. A high-order discontinuous Galerkin method is used to discretize the transformed equation in space and a high-order diagonally implicit Runge-Kutta scheme is used for the temporal discretization. Quantities of interest that take the form of space-time integrals are discretized in a solver-consistent manner. The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law. These quantities of interest and their gradients are used in the context of gradient-based PDE-constrained optimization. The adjoint method is used to solve two optimal shape and control problems governed by the isentropic, compressible Navier-Stokes equations. The first optimization problem seeks the energetically optimal trajectory of a 2D airfoil given a required initial and final spatial position. The optimization solver, driven by gradients computed via the adjoint method, reduced the total energy required to complete the specified mission nearly an order of magnitude. The second optimization problem seeks the energetically optimal flapping motion and time-morphed geometry of a 2D airfoil given an equality constraint on the x-directed impulse generated on the airfoil. The optimization solver satisfied the impulse constraint to greater than 8 digits of accuracy and reduced the required energy between a factor of 2 and 10, depending on the value of the impulse constraint, as compared to the nominal configuration.

  5. Media Appropriation and Explicitation

    Tomas Laurenzo


    Full Text Available This paper presents a novel characterization of new media art together with an exploration of some key aspects of its practice: I propose that new media art’s defining characteristics are media appropriation and explicitation. With media appropriation I refer to the dialectal inscription into the art practice of the knowledge that allows for some particular technological production. I also propose that new media art’s language is constructed in part via the explicitation of certain aspects of more ‘traditional’ art, and that this explicitation allows for a construction of a new vocabulary. Examples of this are the explicitation of randomness, interaction, programming, or of the role that tools and instruments play, among others.

  6. A review of a method for dynamic load distribution, dynamical modeling, and explicit internal force control when two manipulators mutually lift and transport a rigid body object

    Unseren, M.A.


    The paper reviews a method for modeling and controlling two serial link manipulators which mutually lift and transport a rigid body object in a three dimensional workspace. A new vector variable is introduced which parameterizes the internal contact force controlled degrees of freedom. A technique for dynamically distributing the payload between the manipulators is suggested which yields a family of solutions for the contact forces and torques the manipulators impart to the object. A set of rigid body kinematic constraints which restrict the values of the joint velocities of both manipulators is derived. A rigid body dynamical model for the closed chain system is first developed in the joint space. The model is obtained by generalizing the previous methods for deriving the model. The joint velocity and acceleration variables in the model are expressed in terms of independent pseudovariables. The pseudospace model is transformed to obtain reduced order equations of motion and a separate set of equations governing the internal components of the contact forces and torques. A theoretic control architecture is suggested which explicitly decouples the two sets of equations comprising the model. The controller enables the designer to develop independent, non-interacting control laws for the position control and internal force control of the system.

  7. Step-parallel algorithms for stiff initial value problems

    Veen, W.A. van der


    For the parallel integration of stiff initial value problems, three types of parallelism can be employed: 'parallelism across the problem', 'parallelism across the method' and 'parallelism across the steps'. Recently, methods based on Runge-Kutta schemes that use parallelism across the method have b

  8. Special boundedness properties in numerical initial value problems

    Hundsdorfer, W.; Mozartova, A.; Spijker, M.N.


    For Runge-Kutta methods, linear multistep methods and other classes of general linear methods much attention has been paid in the literature to important nonlinear stability properties known as total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize conditions

  9. 具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式%Explicit Square Conserving Schemes of Landau-Lifshitz Equation With Gilbert Component

    孙建强; 马中骐; 秦孟兆



  10. Terminal Convergence Approximation Modified Chebyshev Picard Iteration for Efficient Orbit Propagation

    Probe, A.; Macomber, B.; Kim, D.; Woollands, R.; Junkins, J.


    Modified Chebyshev Picard Iteration (MCPI) is a numerical method for approximating solutions of Ordinary Differential Equations (ODEs). MCPI uses Picard Iteration with Orthogonal Chebyshev Polynomial basis functions to recursively update approximate time histories of system states. Unlike stepping numerical integrators, such as explicit Runge-Kutta methods, MCPI approximates large segments of the trajectory by evaluating the forcing function at multiple nodes along the current approximation during each iteration. Importantly, the Picard sequence theoretically converges to the solution over large time intervals if the forces are continuous and once differentiable. Orthogonality of the basis functions and a vector-matrix formulation allow for low overhead cost, efficient iterations, and parallel evaluation of the forcing function. Despite these advantages MCPI only achieves a geometric rate of convergence. Depending on the quality of the starting approximation, MCPI sometimes requires more function evaluations than competing methods; for parallel applications, this is not a serious drawback, but may be for some serial applications. To improve efficiency, the Terminal Convergence Approximation Modified Chebyshev Picard Iteration (TCA-MCPI) was developed. TCA-MCPI takes advantage of the property that once moderate accuracy of the approximating trajectory has been achieved, the subsequent displacement of nodes asymptotically approaches zero. Applying judicious approximation methods to the force function at each node in the terminal convergence iterations is shown to dramatically reduce the computational cost to achieve accurate convergence. To illustrate this approach we consider high-order spherical-harmonic gravity for high accuracy orbital propagation. When combined with a starting approximation from the 2-body solution TCA-MCPI, is shown to outperform 2 current state-of-practice integration methods for astrodynamics. This paper presents the development of TCA

  11. Stochastic differential equations and a biological system

    Wang, Chunyan


    , Milstein and Runge-Kutta methods are used. Because of the specific feature of the model for the growth process, that its solution does not exist in the general sense, we combine these numerical integration methods with a transformation technique, and the solutions are derived in the Ito sense...

  12. Validation of a simple dynamic thermal performance characterization model based on the piston flow concept for flat-plate solar collectors

    Deng, Jie; Yang, Ming; Ma, Rongjiang


    dynamic model based on the first-order difference method is compared to that of the numerical solution of the collector ordinary differential equation (ODE) model using the fourth-order Runge-Kutta method. The improved thermal inertia model (TIM) on the basis of closed-form solution presented by Deng et...

  13. Upwind scheme for acoustic disturbances generated by low-speed flows

    Ekaterinaris, J.A.


    , compressible how equations, A numerical method for the solution of the equations governing the acoustic field is presented. The primitive variable form of the governing equations is used for the numerical solution. Time integration is performed with a fourth-order, Runge-Kutta method, Discretization...

  14. Linear energy-preserving integrators for Poisson systems

    Cohen, David; Hairer, Ernst


    For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge-Kutta method with infinitely many stages.

  15. On the stability of numerical integration routines.

    Glover, K.; Willems, J. C.


    Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.

  16. On the stability of numerical integration routines for ordinary differential equations.

    Glover, K.; Willems, J. C.


    Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.

  17. The SAMPL5 host-guest challenge: computing binding free energies and enthalpies from explicit solvent simulations by the attach-pull-release (APR) method

    Yin, Jian; Henriksen, Niel M.; Slochower, David R.; Gilson, Michael K.


    The absolute binding free energies and binding enthalpies of twelve host-guest systems in the SAMPL5 blind challenge were computed using our attach-pull-release (APR) approach. This method has previously shown good correlations between experimental and calculated binding data in retrospective studies of cucurbit[7]uril (CB7) and β-cyclodextrin (βCD) systems. In the present work, the computed binding free energies for host octa acid (OA or OAH) and tetra-endo-methyl octa-acid (TEMOA or OAMe) with guests are in good agreement with prospective experimental data, with a coefficient of determination (R2) of 0.8 and root-mean-squared error of 1.7 kcal/mol using the TIP3P water model. The binding enthalpy calculations achieve moderate correlations, with R2 of 0.5 and RMSE of 2.5 kcal/mol, for TIP3P water. Calculations using the newly developed OPC water model also show good performance. Furthermore, the present calculations semi-quantitatively capture the experimental trend of enthalpy-entropy compensation observed, and successfully predict guests with the strongest and weakest binding affinity. The most populated binding poses of all twelve systems, based on clustering analysis of 750 ns molecular dynamics (MD) trajectories, were extracted and analyzed. Computational methods using MD simulations and explicit solvent models in a rigorous statistical thermodynamic framework, like APR, can generate reasonable predictions of binding thermodynamics. Especially with continuing improvement in simulation force fields, such methods hold the promise of making substantial contributions to hit identification and lead optimization in the drug discovery process.

  18. Comparative analysis of solution methods of the punctual kinetic equations; Analisis comparativo de metodos de solucion de las ecuaciones de cinetica puntual

    Hernandez S, A. [UNAM-LAIRN, Jiutepec, Morelos (Mexico)] e-mail:


    The following one written it presents a comparative analysis among different analytical solutions for the punctual kinetics equation, which present two variables of interest: a) the temporary behavior of the neutronic population, and b) The temporary behavior of the different groups of precursors of delayed neutrons. The first solution is based on a method that solves the transfer function of the differential equation for the neutronic population, in which intends to obtain the different poles that give the stability of this transfer function. In this section it is demonstrated that the temporary variation of the reactivity of the system can be managed as it is required, since the integration time for this method doesn't affect the result. However, the second solution is based on an iterative method like that of Runge-Kutta or the Euler method where the algorithm was only used to solve first order differential equations giving this way solution to each differential equation that conforms the equations of punctual kinetics. In this section it is demonstrated that only it can obtain a correct temporary behavior of the neutronic population when it is integrated on an interval of very short time, forcing to the temporary variation of the reactivity to change very quick way without one has some control about the time. In both methods the same change is used so much in the reactivity of the system like in the integration times, giving validity to the results graph the one the temporary behavior of the neutronic population vs. time. (Author)

  19. An Explicit Nonlinear Mapping for Manifold Learning

    Qiao, Hong; Zhang, Peng; Wang, Di; Zhang, Bo


    Manifold learning is a hot research topic in the field of computer science and has many applications in the real world. A main drawback of manifold learning methods is, however, that there is no explicit mappings from the input data manifold to the output embedding. This prohibits the application of manifold learning methods in many practical problems such as classification and target detection. Previously, in order to provide explicit mappings for manifold learning methods, many methods have...

  20. On new solutions of linear system of first -order fuzzy differential equations with fuzzy coefficient

    A. Karimi Dizicheh


    Full Text Available In this paper, we firstly introduce system of first order fuzzy differential equations. Then, we convert the problem to two crisp systems of first order differential equations. For numerical aspects, we apply exponentially fitted Runge Kutta method to solve the fuzzy problems. We solve some well-known examples in order to demonstrate the applicability and accuracy of results.


    Numerical solutions, Runge-Kutta method. Introduction ... the analytical solution of equation of motion of a simple pendulum can only be . - obtained by ..... distribution and the time step (Fig. 7). ,,. .... thank department of Physias for facility disposal. ... Science with Programming and Software Applications. Third ed.

  2. Numerical integration subprogrammes in Fortran II-D

    Fry, C. R.


    This note briefly describes some integration subprogrammes written in FORTRAN II-D for the IBM 1620-II at CARDE. These presented are two Newton-Cotes, Chebyshev polynomial summation, Filon's, Nordsieck's and optimum Runge-Kutta and predictor-corrector methods. A few miscellaneous numerical integration procedures are also mentioned covering statistical functions, oscillating integrands and functions occurring in electrical engineering.

  3. On the accuracy and efficiency of finite difference solutions for nonlinear waves

    Bingham, Harry B.


    -uniform grid. Time-integration is performed using a fourth-order Runge-Kutta scheme. The linear accuracy, stability and convergence properties of the method are analyzed in two-dimensions, and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes...

  4. Numerical Simulation of Piston Ring Lubrication

    Felter, Christian Lotz


    variables u, v, rho for the velocity components and density, respectively. Time integration is performed by a third order Runge-Kutta method. The set of equations is closed by the Dowson- Higginson equation for the relation between density and pressure. Boundary conditions are the non-slip condition...

  5. Aero-Acoustic Computations of Wind Turbines

    Zhu, Wei Jun


    integration, the classical 4-stage Runge-Kutta scheme is applied. Non-centered high-order schemes at numerical boundaries and high-order filter schemes are also discussed due to their importance. The method was validated against a few test cases and further applied for flows around a cylinder and an airfoil...

  6. Hybrid finite-volume-ROM approach to non-linear aerospace fluid-structure interaction modelling

    Mowat, AGB


    Full Text Available frame, describe the fluid domain while the structure is represented by a quadratic modal reduced order model (ROM). A Runge-Kutta dual-timestepping method is employed for the fluid solver, and three upwind schemes are considered viz. AUSM+ -up, HLLC...

  7. Study of the Nonlinear Dropping Shock Response of Expanded Foam Packaging System

    Huan-xin Jiang


    Full Text Available The variational iteration method-2 (VIM-2 is applied to obtain approximate analytical solutions of EPS foam cushioning packaging system. The first-order frequency solution of the equation of motion was obtained and compared with the numerical simulation solution solved by the Runge-Kutta algorithm. The results showed the high accuracy of this VIM with convenient calculation.

  8. Scalar fields in (2+1) dimensions coupled to gravity

    Özçelik, H T; Hortaçsu, M


    We couple a conformal scalar field in (2+1) dimensions to Einstein-Cartan gravity. The field equations are obtained by a variational principle. Einstein-Cartan equations are not solved analytically. These equations are solved numerically with 4th order Runge-Kutta method.

  9. Numerical study of Peakons and k-Solitons of the Camassa-Holm and Holm-Hone equations

    Popov, S. P.


    The spectral Fourier and Runge-Kutta methods are used to study the Camassa-Holm and Holm-Hone equations numerically. Numerical results for problems with initial data leading to the generation and interaction of peakons and k-solitons are discussed.

  10. Theoretical analysis of pulse modulation of semiconductor lasers

    Xu Baoxi; Zhan Yushu; Guo Siji


    Rate equations of Gaussian shape pulse modulated semiconductor lasers are solved by Runge--Kutta method, and the results are analyzed. The formulae for calculating the delay time, pulse width of laser pulse and maximum bit-rate of Gaussian shape pulse modulation are derived. The experimental results of modulation pattern effects are given.

  11. Numerical Solution of Compressible Steady Flows around the RAE 2822 Airfoil

    Kryštůfek, P.; Kozel, K.


    The article presents results of a numerical solution of subsonic, transonic and supersonic flows described by the system of Navier-Stokes equations in 2D laminar compressible flows around the RAE 2822 airfoil. Authors used FVM multistage Runge-Kutta method to numerically solve the flows around the RAE 2822 airfoil.

  12. Numerical simulation of generalized Langevin equation with arbitrary correlated noise.

    Lü, Kun; Bao, Jing-Dong


    A generalized Langevin equation with arbitrary correlated noise and associated frequency-dependent friction is simulated, which can lead to anomalous diffusion. The algorithm is realized by using the Fourier transform technique to generate noise and the stochastic Runge-Kutta method to solve the whole equation. Application to an acoustic phonon model, initial preparation-dependent ballistic diffusion, is shown.

  13. Seismic P-Velocities in Outcrops of the Troodos Ophiolite Complex, Cyprus


    accomplished using RAYINVR, a pub- lic domain program developed at the University of Utah (Zelt and Smith, 1992) and based on a fast raytracing algorith...boundaries. Raytracing within a layer is carried out by solving one of the following sets of equations using a Runge-Kutta method: dz dO (z- vx• ot0

  14. Numerical stability of finite difference algorithms for electrochemical kinetic simulations. Matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods, extended to the 3- and 4-point gradient approximation at the electrodes

    Bieniasz, Leslaw K.; Østerby, Ole; Britz, Dieter


    We extend the analysis of the stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference algorithms for electrochemical kinetic simulations, to the multipoint gradient approximations at the electrode. The discussion is based on the matrix method...... of stability analysis....

  15. An explicit combinatorial design

    Ma, Xiongfeng


    A combinatorial design is a family of sets that are almost disjoint, which is applied in pseudo random number generations and randomness extractions. The parameter, $\\rho$, quantifying the overlap between the sets within the family, is directly related to the length of a random seed needed and the efficiency of an extractor. Nisan and Wigderson proposed an explicit construction of designs in 1994. Later in 2003, Hartman and Raz proved a bound of $\\rho\\le e^2$ for the Nisan-Wigderson construction. In this work, we prove a tighter bound of $\\rhoexplicit weak design with $\\rho=1$.

  16. Building an explicit de Sitter

    Louis, Jan [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Hamburg Univ. (Germany). Zentrum fuer Mathematische Physik; Rummel, Markus; Valandro, Roberto [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Westphal, Alexander [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany). Gruppe Theorie


    We construct an explicit example of a de Sitter vacuum in type IIB string theory that realizes the proposal of Kaehler uplifting. As the large volume limit in this method depends on the rank of the largest condensing gauge group we carry out a scan of gauge group ranks over the Kreuzer-Skarke set of toric Calabi-Yau threefolds. We find large numbers of models with the largest gauge group factor easily exceeding a rank of one hundred. We construct a global model with Kaehler uplifting on a two-parameter model on CP{sup 4}{sub 11169}, by an explicit analysis from both the type IIB and F-theory point of view. The explicitness of the construction lies in the realization of a D7 brane configuration, gauge flux and RR and NS flux choices, such that all known consistency conditions are met and the geometric moduli are stabilized in a metastable de Sitter vacuum with spontaneous GUT scale supersymmetry breaking driven by an F-term of the Kaehler moduli.

  17. An Explicit Nonlinear Mapping for Manifold Learning.

    Qiao, Hong; Zhang, Peng; Wang, Di; Zhang, Bo


    Manifold learning is a hot research topic in the held of computer science and has many applications in the real world. A main drawback of manifold learning methods is, however, that there are no explicit mappings from the input data manifold to the output embedding. This prohibits the application of manifold learning methods in many practical problems such as classification and target detection. Previously, in order to provide explicit mappings for manifold learning methods, many methods have been proposed to get an approximate explicit representation mapping with the assumption that there exists a linear projection between the high-dimensional data samples and their low-dimensional embedding. However, this linearity assumption may be too restrictive. In this paper, an explicit nonlinear mapping is proposed for manifold learning, based on the assumption that there exists a polynomial mapping between the high-dimensional data samples and their low-dimensional representations. As far as we know, this is the hrst time that an explicit nonlinear mapping for manifold learning is given. In particular, we apply this to the method of locally linear embedding and derive an explicit nonlinear manifold learning algorithm, which is named neighborhood preserving polynomial embedding. Experimental results on both synthetic and real-world data show that the proposed mapping is much more effective in preserving the local neighborhood information and the nonlinear geometry of the high-dimensional data samples than previous work.

  18. Explicit correlation factors

    Johnson, Cole M.; Hirata, So; Ten-no, Seiichiro


    We analyze the performance of 17 different correlation factors in explicitly correlated second-order many-body perturbation calculations for correlation energies. Highly performing correlation factors are found to have near-universal shape and size in the short range of electron-electron distance (0 1.5 a.u.) is insignificant insofar as the factor becomes near constant, leaving an orbital expansion to describe decoupled electrons. An analysis based on a low-rank Taylor expansion of the correlation factor seems limited, except that a negative second derivative with the value of around -1.3 a.u. correlates with high performance.

  19. Explicit Time-Stepping for Stiff ODEs

    Eriksson, Kenneth; Logg, Anders; 10.1137/S1064827502409626


    We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvement is large. We illustrate the technique on a number of well-known stiff test problems.

  20. 无阻尼Landau-Lifshitz方程的李群解法%Lie Group Method for Solving Non-damping Landau-Lifshitz Equations

    康永强; 张素英


    文章给出一类求解无阻尼Landau-Lifshitz方程的 Runge-Kutta/Munthe-Kaas方法,属于李群方法,它能保证所得的数值解在系统精确解所在的微分流形上迭代.并讨论了该方法能保持离散系统的二次守恒量.

  1. GPU-Based Parallel Integration of Large Numbers of Independent ODE Systems

    Niemeyer, Kyle E


    The task of integrating a large number of independent ODE systems arises in various scientific and engineering areas. For nonstiff systems, common explicit integration algorithms can be used on GPUs, where individual GPU threads concurrently integrate independent ODEs with different initial conditions or parameters. One example is the fifth-order adaptive Runge-Kutta-Cash-Karp (RKCK) algorithm. In the case of stiff ODEs, standard explicit algorithms require impractically small time-step sizes for stability reasons, and implicit algorithms are therefore commonly used instead to allow larger time steps and reduce the computational expense. However, typical high-order implicit algorithms based on backwards differentiation formulae (e.g., VODE, LSODE) involve complex logical flow that causes severe thread divergence when implemented on GPUs, limiting the performance. Therefore, alternate algorithms are needed. A GPU-based Runge-Kutta-Chebyshev (RKC) algorithm can handle moderate levels of stiffness and performs s...

  2. Real-time study of missiles and rockets simulation based on Runge-Kutta algorithm%基于RK算法的弹箭仿真实时研究

    黄振全; 陈志武



  3. Explicit Spin Coordinates

    Hunter, G; Hunter, Geoffrey; Schlifer, Ian


    The recently established existence of spherical harmonic functions, $Y_\\ell^{m}(\\theta,\\phi)$ for half-odd-integer values of $\\ell$ and $m$, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles $\\theta_s, \\phi_s$, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, $S^2, S_z$, are represented by the usual differential operators in spherical polar coordinates (commonly used for $L^2, L_z$), and their electron-spin eigenfunctions are $\\sqrt{\\sin\\theta_s} \\exp(\\pm\\phi_s/2)$. This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, $\\alpha, \\beta,$ that ``integration over spin coordinates'' is a true integration (over the angles $\\theta_s, \\phi_s$). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an interme...

  4. Topology Optimization using an Explicit Interface Representation

    Christiansen, Asger Nyman; Nobel-Jørgensen, Morten; Bærentzen, J. Andreas;

    Current methods for topology optimization primarily represent the interface between solid and void implicitly on fixed grids. In contrast, shape optimization methods represent the interface explicitly, but do not allow for any topological changes to the structure. Using an explicit interface...... to handle topology changes. It does so by discretizing the entire design domain into an irregular adaptive triangle mesh and thereby explicitly representing both the structure and the embedding space. In other words, the entire design domain is divided into triangles, where the interface is represented...... the final design. The status of the work is that the method has been developed and is showing promising results. For instance, the cantilever beam problem has been solved to a high precision using a fine discretization by evaluating the objective function approximately 500 times. This took around 100...

  5. Using an Explicit Emission Tagging Method in Global Modeling of Source-Receptor Relationships for Black Carbon in the Arctic: Variations, Sources and Transport Pathways

    Wang, Hailong; Rasch, Philip J.; Easter, Richard C.; Singh, Balwinder; Zhang, Rudong; Ma, Po-Lun; Qian, Yun; Ghan, Steven J.; Beagley, Nathaniel


    We introduce an explicit emission tagging technique in the Community Atmosphere Model to quantify source-region-resolved characteristics of black carbon (BC), focusing on the Arctic. Explicit tagging of BC source regions without perturbing the emissions makes it straightforward to establish source-receptor relationships and transport pathways, providing a physically consistent and computationally efficient approach to produce a detailed characterization of the destiny of regional BC emissions and the potential for mitigation actions. Our analysis shows that the contributions of major source regions to the global BC burden are not proportional to the respective emissions due to strong region-dependent removal rates and lifetimes, while the contributions to BC direct radiative forcing show a near-linear dependence on their respective contributions to the burden. Distant sources contribute to BC in remote regions mostly in the mid- and upper troposphere, having much less impact on lower-level concentrations (and deposition) than on burden. Arctic BC concentrations, deposition and source contributions all have strong seasonal variations. Eastern Asia contributes the most to the wintertime Arctic burden. Northern Europe emissions are more important to both surface concentration and deposition in winter than in summer. The largest contribution to Arctic BC in the summer is from Northern Asia. Although local emissions contribute less than 10% to the annual mean BC burden and deposition within the Arctic, the per-emission efficiency is much higher than for major non-Arctic sources. The interannual variability (1996-2005) due to meteorology is small in annual mean BC burden and radiative forcing but is significant in yearly seasonal means over the Arctic. When a slow aging treatment of BC is introduced, the increase of BC lifetime and burden is source-dependent. Global BC forcing-per-burden efficiency also increases primarily due to changes in BC vertical distributions. The

  6. On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations

    Zhang, Xiangxiong


    We construct a local Lax-Friedrichs type positivity-preserving flux for compressible Navier-Stokes equations, which can be easily extended to multiple dimensions for generic forms of equations of state, shear stress tensor and heat flux. With this positivity-preserving flux, any finite volume type schemes including discontinuous Galerkin (DG) schemes with strong stability preserving Runge-Kutta time discretizations satisfy a weak positivity property. With a simple and efficient positivity-preserving limiter, high order explicit Runge-Kutta DG schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy. Numerical tests suggest that the positivity-preserving flux and the positivity-preserving limiter do not induce excessive artificial viscosity, and the high order positivity-preserving DG schemes without other limiters can produce satisfying non-oscillatory solutions when the nonlinear diffusion in compressible Navier-Stokes equations is accurately resolved.

  7. Numerical Stability and Accuracy of Temporally Coupled Multi-Physics Modules in Wind-Turbine CAE Tools

    Gasmi, A.; Sprague, M. A.; Jonkman, J. M.; Jones, W. B.


    In this paper we examine the stability and accuracy of numerical algorithms for coupling time-dependent multi-physics modules relevant to computer-aided engineering (CAE) of wind turbines. This work is motivated by an in-progress major revision of FAST, the National Renewable Energy Laboratory's (NREL's) premier aero-elastic CAE simulation tool. We employ two simple examples as test systems, while algorithm descriptions are kept general. Coupled-system governing equations are framed in monolithic and partitioned representations as differential-algebraic equations. Explicit and implicit loose partition coupling is examined. In explicit coupling, partitions are advanced in time from known information. In implicit coupling, there is dependence on other-partition data at the next time step; coupling is accomplished through a predictor-corrector (PC) approach. Numerical time integration of coupled ordinary-differential equations (ODEs) is accomplished with one of three, fourth-order fixed-time-increment methods: Runge-Kutta (RK), Adams-Bashforth (AB), and Adams-Bashforth-Moulton (ABM). Through numerical experiments it is shown that explicit coupling can be dramatically less stable and less accurate than simulations performed with the monolithic system. However, PC implicit coupling restored stability and fourth-order accuracy for ABM; only second-order accuracy was achieved with RK integration. For systems without constraints, explicit time integration with AB and explicit loose coupling exhibited desired accuracy and stability.


    张士宏; 尚彦凌; 郎利辉; 康达昌; 王仲仁


    The theory and features of the dynamic explicit finite element methods are discussed, the available various commercial FEM codes used for sheet metal forming simulation are introduced. The dynamic explicit FEM code LS-DYNA3D is used for the simulation of a few sheet metal forming processes. Process defects such as wrinkling, local thinning and ruptures are predicted. The design of the tools and the process parameters can thus be improved. The time and the costs of the manufacturing of the tools are reduced, so that it is effective and cost-effective to use the dynamic explicit finite element methods for simulation of sheet metal forming production.%介绍了动态显式有限元法的原理和特点,总结讨论了国际现有各种商用有限元软件的情况。采用显式有限元软件LS-DYNA3D 对板材零件冲压过程进行计算机模拟分析,预测冲压过程中可能出现的各种工艺缺陷,例如坯料的起皱、局部减薄和破裂,并以模拟结果为依据提出改进模具和工艺参数的办法,优化工艺参数,可以减少调试和修模的次数,以此实现降低模具费用、缩短制模时间、提高产品成品率和材料利用率,最终达到减少产品成本的目的。

  9. Isogeometric Collocation for Elastostatics and Explicit Dynamics


    collocation methods within the framework of Isogeometric Analysis (IGA) to multi-patch NURBS con gurations, various boundary and patch interface conditions...development of collocation methods within the framework of Iso- geometric Analysis (IGA) to multi-patch NURBS configurations, various boundary and patch...finite element analysis. Key words: Isogeometric analysis; collocation methods; B-splines; NURBS ; explicit dynamics. 1 Introduction There are many

  10. A Generalized Algebraic Method for Constructing a Series of Explicit Exact Solutions of a (1+1)-Dimensional Dispersive Long Wave Equation

    CHENYong; WANGQi; LIBiao


    Making use of a new and more general ansatz, we present the generalized algebraic method to uniformly construct a series of new and general travelling wave solution for nonlinear partial differential equations. As an application of the method, we choose a (1+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by the method proposed by Fan [E. Fan, Comput. Phys. Commun. 153 (2003) 17] and find other new and more general solutions at the same time, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solutions, hyperbolic and soliton solutions, Jacobi and Weierstrass doubly periodic wave solutions.

  11. A new method for deriving analytical solutions of partial differential equations-Algebraically explicit analytical solutions of two-buoyancy natural convection in porous media


    Analytical solutions of governing equations of various phenomena have their irre-placeable theoretical meanings. In addition, they can also be the benchmark solu-tions to verify the outcomes and codes of numerical solutions, and even to develop various numerical methods such as their differencing schemes and grid generation skills as well. A hybrid method of separating variables for simultaneous partial differential equation sets is presented. It is proposed that different methods of separating variables for different independent variables in the simultaneous equa-tion set may be used to improve the solution derivation procedure, for example, using the ordinary separating method for some variables and using extraordinary methods of separating variables, such as the separating variables with addition promoted by the first author, for some other variables. In order to prove the ability of the above-mentioned hybrid method, a lot of analytical exact solutions of two-buoyancy convection in porous media are successfully derived with such a method. The physical features of these solutions are given.

  12. Matrix Pseudospectral Method for (Visco)Elastic Tides Modeling of Planetary Bodies

    Zabranova, Eliska; Hanyk, Ladidslav; Matyska, Ctirad


    We deal with the equations and boundary conditions describing deformation and gravitational potential of prestressed spherically symmetric elastic bodies by decomposing governing equations into a series of boundary value problems (BVP) for ordinary differential equations (ODE) of the second order. In contrast to traditional Runge-Kutta integration techniques, highly accurate pseudospectral schemes are employed to directly discretize the BVP on Chebyshev grids and a set of linear algebraic equations with an almost block diagonal matrix is derived. As a consequence of keeping the governing ODEs of the second order instead of the usual first-order equations, the resulting algebraic system is half-sized but derivatives of the model parameters are required. Moreover, they can be easily evaluated for models, where structural parametres are piecewise polynomially dependent. Both accuracy and efficiency of the method are tested by evaluating the tidal Love numbers for the Earth's model PREM. Finally, we also derive complex Love numbers for models with the Maxwell viscoelastic rheology, where viscosity is a depth-dependent function. The method is applied to evaluation of the tidal Love numbers for models of Mars and Venus. The Love numbers of the two Martian models - the former optimized to cosmochemical data and the latter to the moment of inertia (Sohl and Spohn, 1997) - are h2=0.172 (0.212) and k2=0.093 (0.113). For Venus, the value of k2=0.295 (Konopliv and Yoder, 1996), obtained from the gravity-field analysis, is consistent with the results for our model with the liquid-core radius of 3110 km (Zábranová et al., 2009). Together with rapid evaluation of free oscillation periods by an analogous method, this combined matrix approach could by employed as an efficient numerical tool in structural studies of planetary bodies. REFERENCES Konopliv, A. S. and Yoder, C. F., 1996. Venusian k2 tidal Love number from Magellan and PVO tracking data, Geophys. Res. Lett., 23, 1857


    Jikun ZHAO; Shaochun CHEN


    In this article,we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method.We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson's element without the regular assumption,respectively,which implies the final finite element error estimate.Such explicit a priori error estimates can be used as computable error bounds.

  14. Explicit Formulas for Meixner Polynomials

    Dmitry V. Kruchinin


    Full Text Available Using notions of composita and composition of generating functions, we show an easy way to obtain explicit formulas for some current polynomials. Particularly, we consider the Meixner polynomials of the first and second kinds.

  15. Chebyshev collocation spectral lattice Boltzmann method for simulation of low-speed flows.

    Hejranfar, Kazem; Hajihassanpour, Mahya


    In this study, the Chebyshev collocation spectral lattice Boltzmann method (CCSLBM) is developed and assessed for the computation of low-speed flows. Both steady and unsteady flows are considered here. The discrete Boltzmann equation with the Bhatnagar-Gross-Krook approximation based on the pressure distribution function is considered and the space discretization is performed by the Chebyshev collocation spectral method to achieve a highly accurate flow solver. To provide accurate unsteady solutions, the time integration of the temporal term in the lattice Boltzmann equation is made by the fourth-order Runge-Kutta scheme. To achieve numerical stability and accuracy, physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used. An iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows. The main advantage of using the CCSLBM over other high-order accurate lattice Boltzmann method (LBM)-based flow solvers is the decay of the error at exponential rather than at polynomial rates. Note also that the CCSLBM applied does not need any numerical dissipation or filtering for the solution to be stable, leading to highly accurate solutions. Three two-dimensional (2D) test cases are simulated herein that are a regularized cavity, the Taylor vortex problem, and doubly periodic shear layers. The results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement. The computational efficiency of the proposed solution methodology based on the CCSLBM is also examined by comparison with those of the standard streaming-collision (classical) LBM and two finite-difference LBM solvers. The study indicates that the CCSLBM provides more accurate and efficient solutions than these LBM solvers in terms of CPU and memory usage and an exponential

  16. Modelling multi-component aerosol transport problems by the efficient splitting characteristic method

    Liang, Dong; Fu, Kai; Wang, Wenqia


    In this paper, a splitting characteristic method is developed for solving general multi-component aerosol transports in atmosphere, which can efficiently compute the aerosol transports by using large time step sizes. The proposed characteristic finite difference method (C-FDM) can solve the multi-component aerosol distributions in high dimensional domains over large ranges of concentrations and for different aerosol types. The C-FDM is first tested to compute the moving of a Gaussian concentration hump. Comparing with the Runge-Kutta method (RKM), our C-FDM can use very large time step sizes. Using Δt = 0.1, the accuracy of our C-FDM is 10-4, but the RKM only gets the accuracy of 10-2 using a small Δt = 0.01 and the accuracy of 10-3 even using a much smaller Δt = 0.002. A simulation of sulfate transport in a varying wind field is then carried out by the splitting C-FDM, where the sulfate pollution is numerically showed expanding along the wind direction and the effects of the different time step sizes and different wind speeds are analyzed. Further, a realistic multi-component aerosol transport over an area in northeastern United States is studied. Concentrations of PM2.5 sulfate, ammonium, nitrate are high in the urban area, and low in the marine area, while sea salts of sodium and chloride mainly exist in the marine area. The normalized mean bias and the normalized mean error of the predicted PM2.5 concentrations are -6.5% and 24.1% compared to the observed data measured at monitor stations. The time series of numerical aerosol concentration distribution show that the strong winds can move the aerosol concentration peaks horizontally for a long distance, such as from the urban area to the rural area and from the marine area to the urban and rural area. Moreover, we also show the numerical time duration patterns of the aerosol concentration distributions due to the affections of the turbulence and the deposition removal. The developed splitting C-FDM algorithm

  17. An immersed boundary method for direct and large eddy simulation of stratified flows in complex geometry

    Rapaka, Narsimha R.; Sarkar, Sutanu


    A sharp-interface Immersed Boundary Method (IBM) is developed to simulate density-stratified turbulent flows in complex geometry using a Cartesian grid. The basic numerical scheme corresponds to a central second-order finite difference method, third-order Runge-Kutta integration in time for the advective terms and an alternating direction implicit (ADI) scheme for the viscous and diffusive terms. The solver developed here allows for both direct numerical simulation (DNS) and large eddy simulation (LES) approaches. Methods to enhance the mass conservation and numerical stability of the solver to simulate high Reynolds number flows are discussed. Convergence with second-order accuracy is demonstrated in flow past a cylinder. The solver is validated against past laboratory and numerical results in flow past a sphere, and in channel flow with and without stratification. Since topographically generated internal waves are believed to result in a substantial fraction of turbulent mixing in the ocean, we are motivated to examine oscillating tidal flow over a triangular obstacle to assess the ability of this computational model to represent nonlinear internal waves and turbulence. Results in laboratory-scale (order of few meters) simulations show that the wave energy flux, mean flow properties and turbulent kinetic energy agree well with our previous results obtained using a body-fitted grid (BFG). The deviation of IBM results from BFG results is found to increase with increasing nonlinearity in the wave field that is associated with either increasing steepness of the topography relative to the internal wave propagation angle or with the amplitude of the oscillatory forcing. LES is performed on a large scale ridge, of the order of few kilometers in length, that has the same geometrical shape and same non-dimensional values for the governing flow and environmental parameters as the laboratory-scale topography, but significantly larger Reynolds number. A non-linear drag law

  18. Asymptotic solution for heat convection-radiation equation

    Mabood, Fazle; Ismail, Ahmad Izani Md [School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang (Malaysia); Khan, Waqar A. [Department of Engineering Sciences, National University of Sciences and Technology, PN Engineering College, Karachi, 75350 (Pakistan)


    In this paper, we employ a new approximate analytical method called the optimal homotopy asymptotic method (OHAM) to solve steady state heat transfer problem in slabs. The heat transfer problem is modeled using nonlinear two-point boundary value problem. Using OHAM, we obtained the approximate analytical solution for dimensionless temperature with different values of a parameter ε. Further, the OHAM results for dimensionless temperature have been presented graphically and in tabular form. Comparison has been provided with existing results from the use of homotopy perturbation method, perturbation method and numerical method. For numerical results, we used Runge-Kutta Fehlberg fourth-fifth order method. It was found that OHAM produces better approximate analytical solutions than those which are obtained by homotopy perturbation and perturbation methods, in the sense of closer agreement with results obtained from the use of Runge-Kutta Fehlberg fourth-fifth order method.

  19. A new particle-like method for high-speed flows with chemical non-equilibrium

    Fábio Rodrigues Guzzo


    Full Text Available The present work is concerned with the numerical simulation of hypersonic blunt body flows with chemical non-equilibrium. New theoretical and numerical formulations for coupling the chemical reaction to the fluid dynamics are presented and validated. The fluid dynamics is defined for a stationary unstructured mesh and the chemical reaction process is defined for “finite quantities” moving through the stationary mesh. The fluid dynamics is modeled by the Euler equations and the chemical reaction rates by the Arrhenius law. Ideal gases are considered. The thermodynamical data are based on JANNAF tables and Burcat’s database. The algorithm proposed by Liou, known as AUSM+, is implemented in a cell-centered based finite volume method and in an unstructured mesh context. Multidimensional limited MUSCL interpolation method is used to perform property reconstructions and to achieve second-order accuracy in space. The minmod limiter is used. The second order accuracy, five stage, Runge-Kutta time-stepping scheme is employed to perform the time march for the fluid dynamics. The numerical code VODE, which is part of the CHEMKIN-II package, is adopted to perform the time integration for the chemical reaction equations. The freestream reacting fluid is composed of H2 and air at the stoichiometric ratio. The emphasis of the present paper is on the description of the new methodology for handling the coupling of chemical and fluid mechanic processes, and its validation by comparison with the standard time-splitting procedure. The configurations considered are the hypersonic flow over a wedge, in which the oblique detonation wave is induced by an oblique shock wave, and the hypersonic flow over a blunt body. Differences between the solutions obtained with each formulation are presented and discussed, including the effects of grid refinement in each case. The primary objective of such comparisons is the validation of the proposed methodology. Moreover, for

  20. Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)

    Robert Nourgaliev; Theo Theofanous; HyeongKae Park; Vincent Mousseau; Dana Knoll


    In recent work (Nourgaliev, Liou, Theofanous, JCP in press) we demonstrated that numerical simulations of interfacial flows in the presence of strong shear must be cast in dynamically sharp terms (sharp interface treatment or SIM), and that moreover they must meet stringent resolution requirements (i.e., resolving the critical layer). The present work is an outgrowth of that work aiming to overcome consequent limitations on the temporal treatment, which become still more severe in the presence of phase change. The key is to avoid operator splitting between interface motion, fluid convection, viscous/heat diffusion and reactions; instead treating all these non-linear operators fully-coupled within a Newton iteration scheme. To this end, the SIM’s cut-cell meshing is combined with the high-orderaccurate implicit Runge-Kutta and the “recovery” Discontinuous Galerkin methods along with a Jacobian-free, Krylov subspace iteration algorithm and its physics-based preconditioning. In particular, the interfacial geometry (i.e., marker’s positions and volumes of cut cells) is a part of the Newton-Krylov solution vector, so that the interface dynamics and fluid motions are fully-(non-linearly)-coupled. We show that our method is: (a) robust (L-stable) and efficient, allowing to step over stability time steps at will while maintaining high-(up to the 5th)-order temporal accuracy; (b) fully conservative, even near multimaterial contacts, without any adverse consequences (pressure/velocity oscillations); and (c) highorder-accurate in spatial discretization (demonstrated here up to the 12th-order for smoothin-the-bulk-fluid flows), capturing interfacial jumps sharply, within one cell. Performance is illustrated with a variety of test problems, including low-Mach-number “manufactured” solutions, shock dynamics/tracking with slow dynamic time scales, and multi-fluid, highspeed shock-tube problems. We briefly discuss preconditioning, and we introduce two physics