Sample records for two-dimensional euler equation


    Li Jiequan; Sheng Wancheng; Zhang Tong; Zheng Yuxi


    In this paper we survey the authors' and related work on two-dimensional Rie-mann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections: 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.

  2. Blow-up conditions for two dimensional modified Euler-Poisson equations

    Lee, Yongki


    The multi-dimensional Euler-Poisson system describes the dynamic behavior of many important physical flows, yet as a hyperbolic system its solution can blow-up for some initial configurations. This article strives to advance our understanding on the critical threshold phenomena through the study of a two-dimensional modified Euler-Poisson system with a modified Riesz transform where the singularity at the origin is removed. We identify upper-thresholds for finite time blow-up of solutions for the modified Euler-Poisson equations with attractive/repulsive forcing.

  3. Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases


    In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for twodimensional flow of Chaplygin gases.

  4. Regularity of Stagnation Point-form Solutions of the Two-dimensional Euler Equations

    Sarria, Alejandro


    A class of semi-bounded solutions of the two-dimensional incompressible Euler equations, satisfying either periodic or Dirichlet boundary conditions, is examined. For smooth initial data, new blowup criteria in terms of the initial concavity profile is presented and the effects that the boundary conditions have on the global regularity of solutions is discussed. In particular, by deriving a formula for a general solution along Lagrangian trajectories, we describe how p...

  5. 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.

  6. A Genuinely Two-Dimensional Scheme for the Compressible Euler Equations

    Sidilkover, David


    We present a new genuinely multidimensional discretization for the compressible Euler equations. It is the only high-resolution scheme known to us where Gauss-Seidel relaxation is stable when applied as a smoother directly to the resulting high-resolution scheme. This allows us to construct a very simple and highly efficient multigrid steady-state solver. The scheme is formulated on triangular (possibly unstructured) meshes.

  7. Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

    Lacave, Christophe; Masmoudi, Nader


    We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size {\\varepsilon} separated by distances {d_{\\varepsilon}} and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of {d_{\\varepsilon}}\\varepsilon} when {\\varepsilon} goes to zero. If {frac{d_{\\varepsilon}}\\varepsilon to infty}, then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, {frac{d_{\\varepsilon}}\\varepsilon to 0}, then the fluid cannot penetrate the porous region, namely, the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of {frac{d_{\\varepsilon}/\\varepsilon^{2+frac1γ}} where {γ in (0,infty]} is related to the geometry of the lateral boundaries of the obstacles. If {d_{\\varepsilon}/\\varepsilon^{2+frac1γ} to infty}, then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions; in particular, it is equal to {\\varepsilon3} for balls.

  8. Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations

    Ibrahim, A. H.; Tiwari, S. N.; Smith, R. E.


    Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

  9. Two-Dimensional Riemann Solver for Euler Equations of Gas Dynamics

    Brio, M.; Zakharian, A. R.; Webb, G. M.


    We construct a Riemann solver based on two-dimensional linear wave contributions to the numerical flux that generalizes the one-dimensional method due to Roe (1981, J. Comput. Phys.43, 157). The solver is based on a multistate Riemann problem and is suitable for arbitrary triangular grids or any other finite volume tessellations of the plane. We present numerical examples illustrating the performance of the method using both first- and second-order-accurate numerical solutions. The numerical flux contributions are due to one-dimensional waves and multidimensional waves originating from the corners of the computational cell. Under appropriate CFL restrictions, the contributions of one-dimensional waves dominate the flux, which explains good performance of dimensionally split solvers in practice. The multidimensional flux corrections increase the accuracy and stability, allowing a larger time step. The improvements are more pronounced on a coarse mesh and for large CFL numbers. For the second-order method, the improvements can be comparable to the improvements resulting from a less diffusive limiter.

  10. Existence of Weak Solutions of Two-dimensional Euler Equations with Initial Vorticity in Lorentz Space L(2,1)(R2)%当初始旋度属于Lornetz空间L(2,1)(R2)时二维Euler方程弱解的存在性



    Some estimates on 2-D Euler equations are given when initial vorticity ω belongs to a Lorentz space L(2,1). Then based on these estimates, it is proved that there exist global weak solutions of two dimensional Euler equations when ω0(2,1)∈L.

  11. A characteristic mapping method for two-dimensional incompressible Euler flows

    Yadav, Badal; Mercier, Olivier; Nave, Jean-Christophe; Schneider, Kai


    We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. The new approach evolves the flow map using the gradient-augmented level set method (GALSM). Since the flow map can be decomposed into submaps (each over a finite time interval), the error can be controlled by choosing the remapping times appropriately. This leads to a numerical scheme that has exponential resolution in linear time. The computational efficiency and the high precision of the method are illustrated for a vortex merger and a four mode flow. Comparisons with a Cauchy-Lagrangian method are also presented. KS thankfully acknowledges financial support from the French Research Federation for Fusion Studies within the framework of the European Fusion Development Agreement (EFDA).

  12. Implicit finite-difference methods for the Euler equations

    Pulliam, T. H.


    The present paper is concerned with two-dimensional Euler equations and with schemes which are in use of the time of this writing. Most of the development presented carries over directly to three dimensions. The characteristics of the two-dimensional Euler equations in Cartesian coordinates are considered along with generalized curvilinear coordinate transformations, metric relations, invariants of the transformation, flux Jacobian matrices and eigensystems, numerical algorithms, flux split algorithms, implicit and explicit nonlinear control (smoothing), upwind differencing in supersonic regions, unsteady and steady-state computation, the diagonal form of implicit algorithm, metric differencing and invariants, boundary conditions, geometry and mesh generation, and sample solutions.

  13. User's Guide for ECAP2D: an Euler Unsteady Aerodynamic and Aeroelastic Analysis Program for Two Dimensional Oscillating Cascades, Version 1.0

    Reddy, T. S. R.


    This guide describes the input data required for using ECAP2D (Euler Cascade Aeroelastic Program-Two Dimensional). ECAP2D can be used for steady or unsteady aerodynamic and aeroelastic analysis of two dimensional cascades. Euler equations are used to obtain aerodynamic forces. The structural dynamic equations are written for a rigid typical section undergoing pitching (torsion) and plunging (bending) motion. The solution methods include harmonic oscillation method, influence coefficient method, pulse response method, and time integration method. For harmonic oscillation method, example inputs and outputs are provided for pitching motion and plunging motion. For the rest of the methods, input and output for pitching motion only are given.

  14. A two-dimensional Euler solution for an unbladed jet engine configuration

    Stewart, Mark E. M.


    A two dimensional, nonaxisymmetric Euler solution in a geometry representative of a jet engine configuration without blades is presented. The domain, including internal and external flow, is covered with a multiblock grid. In order to construct this grid, a domain decomposition technique is used to subdivide the domain, and smooth grids are dimensioned and placed in each block. The Euler solution is verified by examining five theoretical properties. The result demonstrates techniques for performing numerical solutions in complex geometries and provides a foundation for complete engine throughflow calculations.

  15. Numerical study on a canonized Hamiltonian system representing reduced magnetohydrodynamics and its comparison with two-dimensional Euler system

    Kaneko, Yuta


    Introducing a Clebsch-like parameterization, we have formulated a canonical Hamiltonian system on a symplectic leaf of reduced magnetohydrodynamics. An interesting structure of the equations is in that the Lorentz-force, which is a quadratic nonlinear term in the conventional formulation, appears as a linear term -{\\Delta}Q, just representing the current density (Q is a Clebsch variable, and {\\Delta} is the two-dimensional Laplacian); omitting this term reduces the system into the two-dimensional Euler vorticity equation of a neutral fluid. A heuristic estimate shows that current sheets grow exponentially (even in a fully nonlinear regime) together with the action variable P that is conjugate to Q. By numerical simulation, the predicted behavior of the canonical variables, yielding exponential growth of current sheets, has been demonstrated.

  16. ASTROP2-LE: A Mistuned Aeroelastic Analysis System Based on a Two Dimensional Linearized Euler Solver

    Reddy, T. S. R.; Srivastava, R.; Mehmed, Oral


    An aeroelastic analysis system for flutter and forced response analysis of turbomachines based on a two-dimensional linearized unsteady Euler solver has been developed. The ASTROP2 code, an aeroelastic stability analysis program for turbomachinery, was used as a basis for this development. The ASTROP2 code uses strip theory to couple a two dimensional aerodynamic model with a three dimensional structural model. The code was modified to include forced response capability. The formulation was also modified to include aeroelastic analysis with mistuning. A linearized unsteady Euler solver, LINFLX2D is added to model the unsteady aerodynamics in ASTROP2. By calculating the unsteady aerodynamic loads using LINFLX2D, it is possible to include the effects of transonic flow on flutter and forced response in the analysis. The stability is inferred from an eigenvalue analysis. The revised code, ASTROP2-LE for ASTROP2 code using Linearized Euler aerodynamics, is validated by comparing the predictions with those obtained using linear unsteady aerodynamic solutions.

  17. Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body

    Chen, Chao; Du, Lili; Xie, Chunjing; Xin, Zhouping


    The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal velocity in the upstream, there exists a critical value {ρ_cr} such that if the incoming density in the upstream is larger than {ρ_cr}, then there exists a subsonic flow past a wall. Furthermore, {ρ_cr} is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than {ρ_cr}. The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states.


    袁光伟; 沈智军; 闫伟


    In this paper the upwind discontinuous Galerkin methods with triangle meshes for two dimensional neutron transport equations will be studied.The stability for both of the semi-discrete and full-discrete method will be proved.

  19. Approximation of the Long-term Dynamics of the Dynamical System Generated by the Two-dimensional Thermohydraulics Equations

    Tone, Florentina


    Pursuing our work in [18], [17], [20], [5], we consider in this article the two-dimensional thermohydraulics equations. We discretize these equations in time using the implicit Euler scheme and we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

  20. Operator splitting for two-dimensional incompressible fluid equations

    Holden, Helge; Karper, Trygve K


    We analyze splitting algorithms for a class of two-dimensional fluid equations, which includes the incompressible Navier-Stokes equations and the surface quasi-geostrophic equation. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data are sufficiently regular.

  1. Hamiltonian formalism of two-dimensional Vlasov kinetic equation.

    Pavlov, Maxim V


    In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo-Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo-Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.

  2. Control theory based airfoil design using the Euler equations

    Jameson, Antony; Reuther, James


    This paper describes the implementation of optimization techniques based on control theory for airfoil design. In our previous work it was shown that control theory could be employed to devise effective optimization procedures for two-dimensional profiles by using the potential flow equation with either a conformal mapping or a general coordinate system. The goal of our present work is to extend the development to treat the Euler equations in two-dimensions by procedures that can readily be generalized to treat complex shapes in three-dimensions. Therefore, we have developed methods which can address airfoil design through either an analytic mapping or an arbitrary grid perturbation method applied to a finite volume discretization of the Euler equations. Here the control law serves to provide computationally inexpensive gradient information to a standard numerical optimization method. Results are presented for both the inverse problem and drag minimization problem.

  3. Flutter and Forced Response Analyses of Cascades using a Two-Dimensional Linearized Euler Solver

    Reddy, T. S. R.; Srivastava, R.; Mehmed, O.


    Flutter and forced response analyses for a cascade of blades in subsonic and transonic flow is presented. The structural model for each blade is a typical section with bending and torsion degrees of freedom. The unsteady aerodynamic forces due to bending and torsion motions. and due to a vortical gust disturbance are obtained by solving unsteady linearized Euler equations. The unsteady linearized equations are obtained by linearizing the unsteady nonlinear equations about the steady flow. The predicted unsteady aerodynamic forces include the effect of steady aerodynamic loading due to airfoil shape, thickness and angle of attack. The aeroelastic equations are solved in the frequency domain by coupling the un- steady aerodynamic forces to the aeroelastic solver MISER. The present unsteady aerodynamic solver showed good correlation with published results for both flutter and forced response predictions. Further improvements are required to use the unsteady aerodynamic solver in a design cycle.

  4. Control Operator for the Two-Dimensional Energized Wave Equation

    Sunday Augustus REJU


    Full Text Available This paper studies the analytical model for the construction of the two-dimensional Energized wave equation. The control operator is given in term of space and time t independent variables. The integral quadratic objective cost functional is subject to the constraint of two-dimensional Energized diffusion, Heat and a source. The operator that shall be obtained extends the Conjugate Gradient method (ECGM as developed by Hestenes et al (1952, [1]. The new operator enables the computation of the penalty cost, optimal controls and state trajectories of the two-dimensional energized wave equation when apply to the Conjugate Gradient methods in (Waziri & Reju, LEJPT & LJS, Issues 9, 2006, [2-4] to appear in this series.

  5. Numerical blowup in two-dimensional Boussinesq equations

    Yin, Zhaohua


    In this paper, we perform a three-stage numerical relay to investigate the finite time singularity in the two-dimensional Boussinesq approximation equations. The initial asymmetric condition is the middle-stage output of a $2048^2$ run, the highest resolution in our study is $40960^2$, and some signals of numerical blowup are observed.

  6. Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

    Naso, A; Dubrulle, B


    A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. The vorticity fluctuations are Gaussian while the mean flow is characterized by a linear $\\overline{\\omega}-\\psi$ relationship. Furthermore, the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. A new class of relaxation equations towards the statistical equilibrium state is derived. These equations can provide an effective description of the dynamics towards equilibrium or serve as numerical algorithms to determin...

  7. Euler's Amazing Way to Solve Equations.

    Flusser, Peter


    Presented is a series of examples that illustrate a method of solving equations developed by Leonhard Euler based on an unsubstantiated assumption. The method integrates aspects of recursion relations and sequences of converging ratios and can be extended to polynomial equation with infinite exponents. (MDH)

  8. Lyapunov Computational Method for Two-Dimensional Boussinesq Equation

    Mabrouk, Anouar Ben


    A numerical method is developed leading to Lyapunov operators to approximate the solution of two-dimensional Boussinesq equation. It consists of an order reduction method and a finite difference discretization. It is proved to be uniquely solvable and analyzed for local truncation error for consistency. The stability is checked by using Lyapunov criterion and the convergence is studied. Some numerical implementations are provided at the end of the paper to validate the theoretical results.

  9. Small global solutions to the damped two-dimensional Boussinesq equations

    Adhikari, Dhanapati; Cao, Chongsheng; Wu, Jiahong; Xu, Xiaojing

    The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the “vortex stretching”, we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption. By positioning the solutions in a suitable functional setting (more precisely, the homogeneous Besov space B˚∞,11), we are able to obtain a unique global solution under a minimal smallness assumption.

  10. Remark on Single Exponential Bound of the Vorticity Gradient for the Two-Dimensional Euler Flow Around a Corner

    Itoh, Tsubasa; Miura, Hideyuki; Yoneda, Tsuyoshi


    In this paper, we consider the two-dimensional Euler flow under a simple symmetry condition, with hyperbolic structure in a unit square {D = {(x_1,x_2):0 < x_1+x_2 < √{2},0 < -x_1+x_2 < √{2}}}. It is shown that the Lipschitz estimate of the vorticity on the boundary is at most a single exponential growth near the stagnation point.

  11. Financial integration in Europe : Evidence from Euler equation tests

    Lemmen, J.J.G.; Eijffinger, S.C.W.


    This paper applies Obstfeld's Euler equation tests to assess the degree of financial integration in the European Union. In addition, we design a new Euler equation test which is intimately related to Obstfeld's Euler equation tests. Using data from the latest Penn World Table (Mark 6), we arrive at

  12. Stabilities for nonisentropic Euler-Poisson equations.

    Cheung, Ka Luen; Wong, Sen


    We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repulsive forces.

  13. Solving Nonlinear Euler Equations with Arbitrary Accuracy

    Dyson, Rodger W.


    A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.

  14. The Hamiltonian Canonical Form for Euler-Lagrange Equations

    ZHENG Yu


    Based on the theory of calculus of variation, some suffcient conditions are given for some Euler-Lagrangcequations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile,some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler Lagrangeequation and the cylindric shell equations are given.

  15. Equation of State of the Two-Dimensional Hubbard Model

    Cocchi, Eugenio; Miller, Luke A.; Drewes, Jan H.; Koschorreck, Marco; Pertot, Daniel; Brennecke, Ferdinand; Köhl, Michael


    The subtle interplay between kinetic energy, interactions, and dimensionality challenges our comprehension of strongly correlated physics observed, for example, in the solid state. In this quest, the Hubbard model has emerged as a conceptually simple, yet rich model describing such physics. Here we present an experimental determination of the equation of state of the repulsive two-dimensional Hubbard model over a broad range of interactions 0 ≲U /t ≲20 and temperatures, down to kBT /t =0.63 (2 ) using high-resolution imaging of ultracold fermionic atoms in optical lattices. We show density profiles, compressibilities, and double occupancies over the whole doping range, and, hence, our results constitute benchmarks for state-of-the-art theoretical approaches.

  16. Prandtl's Boundary Layer Equation for Two-Dimensional Flow: Exact Solutions via the Simplest Equation Method

    Taha Aziz


    Full Text Available The simplest equation method is employed to construct some new exact closed-form solutions of the general Prandtl's boundary layer equation for two-dimensional flow with vanishing or uniform mainstream velocity. We obtain solutions for the case when the simplest equation is the Bernoulli equation or the Riccati equation. Prandtl's boundary layer equation arises in the study of various physical models of fluid dynamics. Thus finding the exact solutions of this equation is of great importance and interest.

  17. Convergence of Compressible Euler-Maxwell Equations to Compressible Euler-Poisson Equations


    In this paper, the convergence of time-dependent Euler-Maxwell equations to compressible Euler-Poisson equations in a torus via the non-relativistic limit is studied.The local existence of smooth solutions to both systems is proved by using energy estimates for first order symmetrizable hyperbolic systems. For well prepared initial data the convergence of solutions is rigorously justified by an analysis of asymptotic expansions up to any order. The authors perform also an initial layer analysis for general initial data and prove the convergence of asymptotic expansions up to first order.


    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.


    SHI Wei-hui; WANG Yue-peng; SHEN Chun


    The stability about Navier-Stokes equation and Euler equation was brought into comparison. And by taking their typical initial value problem for example, the reason of leading to the difference in stability between Navier-Stokes equation and Euler equation was also analyzed.

  20. Two Dimensional Tensor Product B-Spline Wavelet Scaling Functions for the Solution of Two-Dimensional Unsteady Diffusion Equations

    XIONG Lei; LI haijiao; ZHANG Lewen


    The fourth-order B spline wavelet scaling functions are used to solve the two-dimensional unsteady diffusion equation. The calculations from a case history indicate that the method provides high accuracy and the computational efficiency is enhanced due to the small matrix derived from this method.The respective features of 3-spline wavelet scaling functions, 4-spline wavelet scaling functions and quasi-wavelet used to solve the two-dimensional unsteady diffusion equation are compared. The proposed method has potential applications in many fields including marine science.

  1. Potential singularity mechanism for the Euler equations

    Brenner, Michael P.; Hormoz, Sahand; Pumir, Alain


    Singular solutions to the Euler equations could provide essential insight into the formation of very small scales in highly turbulent flows. Previous attempts to find singular flow structures have proven inconclusive. We reconsider the problem of interacting vortex tubes, for which it has long been observed that the flattening of the vortices inhibits sustained self-amplification of velocity gradients. Here we consider an iterative mechanism, based on the transformation of vortex filaments into sheets and their subsequent instability back into filaments. Elementary fluid mechanical arguments are provided to support the formation of a singular structure via this iterated mechanism, which we analyze based on a simplified model of filament interactions.

  2. Multiblock, Multigrid Solution Of Euler Equations

    Melson, N. Duane; Cannizzaro, Frank E.; Von Lavante, E.


    Method of numerical solution of Euler equations of three-dimensional flow of compressible fluid involves combination of multiblock and multigrid strategies. In multiblock strategy, flow field divided, into multiple smaller, more computationally-convenient zones and computational grid fitted to applicable flow boundaries generated in each block. In multigrid strategy used here, different quantities computed, variously, on finer or coarser grids. Minimizing cost of computation by using fewest grid points yielding acceptably accurate values of affected variable. Multigrid strategy found effective in accelerating convergence to steady state, while multiblock strategy provides geometric flexibility.

  3. Regularity and Energy Conservation for the Compressible Euler Equations

    Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Wiedemann, Emil


    We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by Constantin et al. for the homogeneous incompressible Euler equations.

  4. Genuinely Multidimensional Kinetic Scheme For Euler Equations

    Tiwari, Praveer


    A new framework based on Boltzmann equation which is genuinely multidimensional and mesh-less is developed for solving Euler's equations. The idea is to use the method of moment of Boltzmann equation to operate in multidimensions using polar coordinates. The aim is to develop a framework which is genuinely multidimensional and can be implemented with different methodologies, no matter whether it is in finite difference, finite volume or finite element form. There is a considerable improvement in capturing shocks and other discontinuities. Also, since the method is multidimensional, the flow features are captured isotropically. The method is further extended to second order using 'Arc of Approach' concept. The framework is developed as a finite difference method (called as GINEUS) and is tested on the benchmark test cases. The results are compared against Kinetic Flux Vector Splitting Method.

  5. The Chandrasekhar's Equation for Two-Dimensional Hypothetical White Dwarfs

    De, Sanchari


    In this article we have extended the original work of Chandrasekhar on the structure of white dwarfs to the two-dimensional case. Although such two-dimensional stellar objects are hypothetical in nature, we strongly believe that the work presented in this article may be prescribed as Master of Science level class problem for the students in physics.

  6. Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

    Chunrong Zhu


    Full Text Available In this paper, we develop the symmetry-related methods to study invariant subspaces of the two-dimensional nonlinear differential operators. The conditional Lie–Bäcklund symmetry and Lie point symmetry methods are used to construct invariant subspaces of two-dimensional differential operators. We first apply the multiple conditional Lie–Bäcklund symmetries to derive invariant subspaces of the two-dimensional operators. As an application, the invariant subspaces for a class of two-dimensional nonlinear quadratic operators are provided. Furthermore, the invariant subspace method in one-dimensional space combined with the Lie symmetry reduction method and the change of variables is used to obtain invariant subspaces of the two-dimensional nonlinear operators.

  7. High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves

    Christiansen, Torben Robert Bilgrav; Bingham, Harry B.; Engsig-Karup, Allan Peter


    The incompressible Euler equations are solved with a free surface, the position of which is captured by applying an Eulerian kinematic boundary condition. The solution strategy follows that of [1, 2], applying a coordinate-transformation to obtain a time-constant spatial computational domain which...... with a two-dimensional implementation of the model are compared with highly accurate stream function solutions to the nonlinear wave problem, which show the approximately expected convergence rates and a clear advantage of using high-order finite difference schemes in combination with the Euler equations....

  8. Determination of two-dimensional magnetostatic equilibria and analogous Euler flows

    Linardatos, D.


    A modified computational procedure with an improved time-stepping algorithm for two-dimensional magnetic relaxation is developed. The procedure is used to determine a family of flows in a closed (square) domain with a single elliptic stagnation point. In addition, the problem of saddle point collapse is investigated, and the tendency to form discontinuities is confirmed in the manner described by Bajer (1989).

  9. Incompressible limit of solutions of multidimensional steady compressible Euler equations

    Chen, Gui-Qiang G.; Huang, Feimin; Wang, Tian-Yi; Xiang, Wei


    A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.

  10. Energy equation, the dissipation function and the Euler turbine equation

    Mobarak, A. (Cairo Univ. (Egypt). Faculty of Engineering)


    The derivation of the energy equation for a rotating frame of coordinates is presented. The link between the thermodynamics and the fluid dynamics of viscous flow and which is generally given by the dissipation function is discussed in more detail. This work shows, that the published definition of the dissipation function is an improper one, and leads in connection with the energy equation to contradictory results when considering the principle of energy conservation. Further, the Euler turbine equation is discussed, and it is shown that the present form is only valid, if the flow condition in the rotor (the relative system) is steady.

  11. Stochastic Euler Equations of Fluid Dynamics with Lvy Noise


    Asymptotic Analysis 99 (2016) 67–103 67 DOI 10.3233/ASY-161376 IOS Press Stochastic Euler equations of fluid dynamics with Lévy noise Manil T. Mohan...References [1] D. Applebaum, Lévy Processes and Stochastic Calculus , Cambridge Studies in Advanced Mathematics, Vol. 93, Cam- bridge University Press...2004. [2] H. Bessaih and F. Flandoli, 2-D Euler equation perturbed by noise, Nonlinear Differential Equations and Applications 6 (1998), 35–54. doi




    The nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations are proposed. The existence and uniqueness of global generalized solution of these equations,and the convergence of approximate solutions are also obtained.

  13. Numerical investigations on the finite time singularity in two-dimensional Boussinesq equations

    Yin, Z


    To investigate the finite time singularity in three-dimensional (3D) Euler flows, the simplified model of 3D axisymmetric incompressible fluids (i.e., two-dimensional Boussinesq approximation equations) is studied numerically. The system describes a cap-like hot zone of fluid rising from the bottom, while the edges of the cap lag behind, forming eye-like vortices. The hot liquid is driven by the buoyancy and meanwhile attracted by the vortices, which leads to the singularity-forming mechanism in our simulation. In the previous 2D Boussinesq simulations, the symmetricial initial data is used. However, it is observed that the adoption of symmetry leads to coordinate singularity. Moreover, as demonstrated in this work that the locations of peak values for the vorticity and the temperature gradient becomes far apart as $t$ approaches the predicted blow-up time. This suggests that the symmetry assumption may be unreasonable for searching solution blow-ups. One of the main contributions of this work is to propose a...

  14. Remarks on the improved regularity criterion for the 2D Euler-Boussinesq equations with supercritical dissipation

    Ye, Zhuan


    This paper is devoted to the investigation of the regularity criterion to the two-dimensional (2D) Euler-Boussinesq equations with supercritical dissipation. By making use of the Littlewood-Paley technique, we provide an improved regularity criterion involving the temperature at the scaling invariant level, which improves the previous results.

  15. A Finite Volume Backward Euler Difference Method for Nonlinear Parabolic Integral-differential Equation%非线性抛物型积分-微分方程的向后Euler差分有限体积元方法

    王波; 王强


    The Finite volume backward Euler difference method is established to discuss two-dimensional parabolic integro-differential equations.These results are new for finite volume element methods for parabolic integro-differential equations.

  16. Certain theorems on two dimensional Laplace transform and non-homogeneous parabolic partial differential equations

    A. Aghili


    Full Text Available In this work,we present new theorems on two-dimensional Laplace transformation. We also develop some applications based on these results. The two-dimensional Laplace transformation is useful in the solution of non-homogeneous partial differential equations. In the last section a boundary value problem is solved by using the double Laplace-Carson transform.

  17. Generalized Burgers equations and Euler-Painlevé transcendents. III

    Sachdev, P. L.; Nair, K. R. C.; Tikekar, V. G.


    It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986); P. L. Sachdev and K. R. C. Nair, ibid. 28, 977 (1987)] that the Euler-Painlevé equations y(d2y/dη2)+a(dy/dη)2 +f(η)y(dy/dη)+g(η)y2+b(dy/dη) +c=0 represent generalized Burgers equations (GBE's) in the same way as Painlevé equations represent the Korteweg-de Vries type of equations. The earlier studies were carried out in the context of GBE's with damping and those with spherical and cylindrical symmetry. In the present paper, GBE's with variable coefficients of viscosity and those with inhomogeneous terms are considered for their possible connection to Euler-Painlevé equations. It is found that the Euler-Painlevé equation, which represents the GBE ut+uβux=(δ/2)g(t)uxx, g(t)=(1+t)n, β>0, has solutions, which either decay or oscillate at η=±∞, only when -1Kamke [Differential Gleichungen : Lösungsmethoden und Lösungen (Akademische Verlagsgesellschaft, Leipzig, 1943)] and Murphy [Ordinary Differential Equations and their Solutions (Van Nostrand, Princeton, NJ, 1960)]. These latter equations arise from a wide range of physical applications and are of some historical interest as well. They are all special cases of a slightly generalized form of the Euler-Painlevé equation.

  18. Euler and the Ordinary Differential Equations

    Taborda, Jonathan


    The following notes are intended to make a small digression on the topics mentioned in the title of the same, since these were not addressed in the past tribute by the Institute of Physics of the UdeA. We believe more than platitude try to justify the importance and effectiveness in the development of mathematics and physics during the eighteenth century and present such issues, therefore a brief description of the methods and problems attacked by Euler and his contemporaries using the heuristics. Note in advance that they constitute a strong impoverished attempt to honor the memory of who is considered the Shakespeare of Mathematics: Universal, rich in detail and inexhaustible.



    The authors establish the existence of a large class of mathematical entropies (the so-called weak entropies) associated with the Euler equations for an isentropic, compressible fluid governed by a general pressure law. A mild assumption on the behavior of the pressure law near the vacuum is solely required. The analysis is based on an asymptotic expansion of the fundamental solution (called here the entropy kernel) of a highly singular Euler-Poisson-Darboux equation. The entropy kernel is only H lder continuous and its regularity is carefully investigated. Relying on a notion introduced earlier by the authors, it is also proven that, for the Euler equations, the set of entropy flux-splittings coincides with the set of entropies-entropy fluxes. These results imply the existence of a flux-splitting consistent with all of the entropy inequalities.

  20. A genuinely multidimensional upwind scheme and efficient multigrid solver for the compressible Euler equations

    Sidilkover, David


    We present a new approach towards the construction of a genuinely multidimensional high-resolution scheme for computing steady-state solutions of the Euler equations of gas dynamics. The unique advantage of this approach is that the Gauss-Seidel relaxation is stable when applied directly to the high-resolution discrete equations, thus allowing us to construct a very efficient and simple multigrid steady-state solver. This is the only high-resolution scheme known to us that has this property. The two-dimensional scheme is presented in detail. It is formulated on triangular (structured and unstructured) meshes and can be interpreted as a genuinely two-dimensional extension of the Roe scheme. The quality of the solutions obtained using this scheme and the performance of the multigrid algorithm are illustrated by the numerical experiments. Construction of the three dimensional scheme is outlined briefly as well.

  1. On the Equivalence of Euler-Lagrange and Noether Equations

    Faliagas, A. C., E-mail: [University of Athens, Department of Mathematics (Greece)


    We prove that, under the condition of nontriviality, the Euler-Lagrange and Noether equations are equivalent for a general class of scalar variational problems. Examples are position independent Lagrangians, Lagrangians of p-Laplacian type, and Lagrangians leading to nonlinear Poisson equations. As applications we prove certain propositions concerning the nonlinear Poisson equation and its generalisations, and the equivalence of admissible and inner variations for the systems under consideration.

  2. Solution of two-dimensional Fredholm integral equation via RBF-triangular method

    Amir Fallahzadeh


    Full Text Available In this paper, a new method is introduced to solve a two-dimensional Fredholm integral equation. The method is based on the approximation by Gaussian radial basis functions and triangular nodes and weights. Also, a new quadrature is introduced to approximate the two dimensional integrals which is called the triangular method. The results of the example illustrate the accuracy of the proposed method increases.

  3. Reduction of Multidimensional Wave Equations to Two-Dimensional Equations: Investigation of Possible Reduced Equations

    Yehorchenko, Irina


    We study possible Lie and non-classical reductions of multidimensional wave equations and the special classes of possible reduced equations - their symmetries and equivalence classes. Such investigation allows to find many new conditional and hidden symmetries of the original equations.

  4. Asymptotic Behavior of the Newton-Boussinesq Equation in a Two-Dimensional Channel

    Fucci, Guglielmo; Singh, Preeti


    We prove the existence of a global attractor for the Newton-Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor.




    This paper studies a two-dimensional modified Navier-stokes equations. The author shows the existence and uniqueness of weak solutions for this equation by Galerkin method in bounded domains. The result is further extended to the case of unbounded channel-like domains.

  6. Integrability of Nonlinear Equations of Motion on Two-Dimensional World Sheet Space-Time

    YAN Jun


    The integrability character of nonlinear equations of motion of two-dimensional gravity with dynamical torsion and bosonic string coupling is studied in this paper. The space-like and time-like first integrals of equations of motion are also found.

  7. Non-uniqueness for the Euler equations: the effect of the boundary

    Bardos, Claude; Wiedemann, Emil


    We consider rotational initial data for the two-dimensional incompressible Euler equations on an annulus. Using the convex integration framework, we show that there exist infinitely many admissible weak solutions (i.e. such with non-increasing energy) for such initial data. As a consequence, on bounded domains there exist admissible weak solutions which are not dissipative in the sense of P.-L. Lions, as opposed to the case without physical boundaries. Moreover we show that admissible solutions are dissipative provided they are H\\"{o}lder continuous near the boundary of the domain.

  8. Non-uniqueness for the Euler equations: the effect of the boundary

    Bardos, C.; Szekelyhidi, L., Jr.; Wiedemann, E.


    Rotational initial data is considered for the two-dimensional incompressible Euler equations on an annulus. With use of the convex integration framework it is shown that there exist infinitely many admissible weak solutions (that is, with non-increasing energy) for such initial data. As a consequence, on bounded domains there exist admissible weak solutions which are not dissipative in the sense of Lions, as opposed to the case without physical boundaries. Moreover, it is shown that admissible solutions are dissipative if they are Hölder continuous near the boundary of the domain. Bibliography: 34 titles.

  9. The Radially Symmetric Euler Equations as an Exterior Differential System

    Baty, Roy; Ramsey, Scott; Schmidt, Joseph


    This work develops the Euler equations as an exterior differential system in radially symmetric coordinates. The Euler equations are studied for unsteady, compressible, inviscid fluids in one-dimensional, converging flow fields with a general equation of state. The basic geometrical constructions (for example, the differential forms, tangent planes, jet space, and differential ideal) used to define and analyze differential equations as systems of exterior forms are reviewed and discussed for converging flows. Application of the Frobenius theorem to the question of the existence of solutions to radially symmetric converging flows is also reviewed and discussed. The exterior differential system is further applied to derive and analyze the general family of characteristic vector fields associated with the one-dimensional inviscid flow equations.

  10. A note on singularities of the 3-D Euler equation

    Tanveer, S.


    In this paper, we consider analytic initial conditions with finite energy, whose complex spatial continuation is a superposition of a smooth background flow and a singular field. Through explicit calculation in the complex plane, we show that under some assumptions, the solution to the 3-D Euler equation ceases to be analytic in the real domain in finite time.

  11. An improved front tracking method for the Euler equations

    Witteveen, J.A.S.; Koren, B.; Bakker, P.G.


    An improved front tracking method for hyperbolic conservation laws is presented. The improved method accurately resolves discontinuities as well as continuous phenomena. The method is based on an improved front interaction model for a physically more accurate modeling of the Euler equations, as comp

  12. A Historical Discussion of Angular Momentum and its Euler Equation

    Sparavigna, Amelia Carolina


    We propose a discussion of angular momentum and its Euler equation, with the aim of giving a short outline of their history. This outline can be useful for teaching purposes too, to amend some problems that students can have in learning this important physical quantity.

  13. State vector splitting for the Euler equations of gasdynamics

    Öksüzoglu, H.


    A new upwind scheme is introduced for the Euler equations of gasdynamics in multi- dimensions. Its relation to Steger-Warming Flux Vector Splitting is discussed. Imple- mentation of the conservative boundary condi- tions on solid walls is also given. The method is intuitive, easy to implement and do

  14. Towards Perfectly Absorbing Boundary Conditions for Euler Equations

    Hayder, M. Ehtesham; Hu, Fang Q.; Hussaini, M. Yousuff


    In this paper, we examine the effectiveness of absorbing layers as non-reflecting computational boundaries for the Euler equations. The absorbing-layer equations are simply obtained by splitting the governing equations in the coordinate directions and introducing absorption coefficients in each split equation. This methodology is similar to that used by Berenger for the numerical solutions of Maxwell's equations. Specifically, we apply this methodology to three physical problems shock-vortex interactions, a plane free shear flow and an axisymmetric jet- with emphasis on acoustic wave propagation. Our numerical results indicate that the use of absorbing layers effectively minimizes numerical reflection in all three problems considered.

  15. Weyl-Euler-Lagrange Equations of Motion on Flat Manifold

    Zeki Kasap


    Full Text Available This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented.

  16. Textbook Multigrid Efficiency for the Steady Euler Equations

    Roberts, Thomas W.; Sidilkover, David; Swanson, R. C.


    A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Computations for channel flow and flow over a nonlifting airfoil have computed. Using Gauss-Seidel relaxation ordered in the flow direction, textbook multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier-Stokes equations.

  17. Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity

    Dijkgraaf, R.; Verlinde, H. (Princeton Univ., NJ (USA). Joseph Henry Labs.); Verlinde, E. (Institute for Advanced Study, Princeton, NJ (USA). School of Natural Sciences)


    We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models. (orig.).

  18. Loop Equations and Virasoro Constraints in Non-Perturbative Two-Dimensional Quantum Gravity

    Dijkgraaf, Robbert; Verlinde, Herman; Verlinde, Erik

    We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.

  19. Exact solutions and conservation laws of the system of two-dimensional viscous Burgers equations

    Abdulwahhab, Muhammad Alim


    Fluid turbulence is one of the phenomena that has been studied extensively for many decades. Due to its huge practical importance in fluid dynamics, various models have been developed to capture both the indispensable physical quality and the mathematical structure of turbulent fluid flow. Among the prominent equations used for gaining in-depth insight of fluid turbulence is the two-dimensional Burgers equations. Its solutions have been studied by researchers through various methods, most of which are numerical. Being a simplified form of the two-dimensional Navier-Stokes equations and its wide range of applicability in various fields of science and engineering, development of computationally efficient methods for the solution of the two-dimensional Burgers equations is still an active field of research. In this study, Lie symmetry method is used to perform detailed analysis on the system of two-dimensional Burgers equations. Optimal system of one-dimensional subalgebras up to conjugacy is derived and used to obtain distinct exact solutions. These solutions not only help in understanding the physical effects of the model problem but also, can serve as benchmarks for constructing algorithms and validation of numerical solutions of the system of Burgers equations under consideration at finite Reynolds numbers. Independent and nontrivial conserved vectors are also constructed.

  20. Fluctuations of large-scale jets in the stochastic 2D Euler equation

    Nardini, Cesare


    Two-dimensional turbulence in a rectangular domain self-organises into large-scale unidirectional jets. While several results are present to characterize the mean jets velocity profile, much less is known about the fluctuations. We study jets dynamics in the stochastically forced two-dimensional Euler equations. In the limit where the average jets velocity profile evolves slowly with respect to turbulent fluctuations, we employ a multi-scale (kinetic theory) approach, which relates jet dynamics to the statistics of Reynolds stresses. We study analytically the Gaussian fluctuations of Reynolds stresses and predict the spatial structure of the jets velocity covariance. Our results agree qualitatively well with direct numerical simulations, clearly showing that the jets velocity profile are enhanced away from the stationary points of the average velocity profile. A numerical test of our predictions at quantitative level seems out of reach at the present day.

  1. Solution of the two- dimensional heat equation for a rectangular plate



    Full Text Available Laplace equation is a fundamental equation of applied mathematics. Important phenomena in engineering and physics, such as steady-state temperature distribution, electrostatic potential and fluid flow, are modeled by means of this equation. The Laplace equation which satisfies boundary values is known as the Dirichlet problem. The solutions to the Dirichlet problem form one of the most celebrated topics in the area of applied mathematics. In this study, a novel method is presented for the solution of two-dimensional heat equation for a rectangular plate. In this alternative method, the solution function of the problem is based on the Green function, and therefore on elliptic functions.

  2. Solution of the two-dimensional compressible Navier-Stokes equations on embedded structured multiblock meshes

    Szmelter, J.; Marchant, M. J.; Evans, A.; Weatherill, N. P.

    A cell vertex finite volume Jameson scheme is used to solve the 2D compressible, laminar, viscous fluid flow equations on locally embedded multiblock meshes. The proposed algorithm is applicable to both the Euler and Navier-Stokes equations. It is concluded that the adaptivity method is very successful in efficiently improving the accuracy of the solution. Both the mesh generator and the flow equation solver which are based on a quadtree data structure offer good flexibility in the treatment of interfaces. It is concluded that methods under consideration lead to accurate flow solutions.

  3. Analytic Solution for Two-Dimensional Heat Equation for an Ellipse Region

    Nurcan Baykus Savasaneril


    Full Text Available In this study, an altenative method is presented for the solution of two-dimensional heat equation in an ellipse region. In this method, the solution function of the problem is based on the Green, and therefore on elliptic functions. To do this, it is made use of the basic consepts associated with elliptic integrals, conformal mappings and Green functions.

  4. Two-dimensional trace-normed canonical systems of differential equations and selfadjoint interface conditions

    de Snoo, H; Winkler, Henrik


    The class of two-dimensional trace-normed canonical systems of differential equations on R is considered with selfadjoint interface conditions at 0. If one or both of the intervals around 0 are H-indivisible the interface conditions which give rise to selfadjoint relations (multi-valued operators) a

  5. The transfer function analysis of various schemes for the two-dimensional shallow-water equations

    Neta, B.; DeVito, C.L.


    In this paper various finite difference and finite element approximations to the linearized two-dimensional shallow-water equations are analyzed. This analysis complements previous results for the one-dimensional case. The first author would like to thank the NPS Foundation Research program for its support of this research.

  6. Study of the forward Dirichlet boundary value problem for the two-dimensional Electrical Impedance Equation

    T, M P Ramirez


    Using a conjecture that allows to approach separable-variables conductivity functions, the elements of the Modern Pseudoanalytic Function Theory are used, for the first time, to numerically solve the Dirichlet boundary value problem of the two-dimensional Electrical Impedance Equation, when the conductivity function arises from geometrical figures, located within bounded domains.

  7. Bound states of two-dimensional Schr\\"{o}dinger-Newton equations

    Stubbe, Joachim


    We prove an existence and uniqueness result for ground states and for purely angular excitations of two-dimensional Schr\\"{o}dinger-Newton equations. From the minimization problem for ground states we obtain a sharp version of a logarithmic Hardy-Littlewood-Sobolev type inequality.

  8. Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

    Reza Abazari


    Full Text Available The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM, and compared with the differential transform method (DTM. The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences.

  9. Optimal Control Strategies in a Two Dimensional Differential Game Using Linear Equation under a Perturbed System

    Musa Danjuma SHEHU


    Full Text Available This paper lays emphasis on formulation of two dimensional differential games via optimal control theory and consideration of control systems whose dynamics is described by a system of Ordinary Differential equation in the form of linear equation under the influence of two controls U(. and V(.. Base on this, strategies were constructed. Hence we determine the optimal strategy for a control say U(. under a perturbation generated by the second control V(. within a given manifold M.

  10. Group classification of steady two-dimensional boundary-layer stagnation-point flow equations

    Nadjafikhah, Mehdi; Hejazi, Seyed Reza


    Lie symmetry group method is applied to study the boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.

  11. Numerical solution of a class of nonlinear two-dimensional integral equations using Bernoulli polynomials

    Sohrab Bazm


    Full Text Available In this study, the Bernoulli polynomials are used to obtain an approximate solution of a class of nonlinear two-dimensional integral equations. To this aim, the operational matrices of integration and the product for Bernoulli polynomials are derived and utilized to reduce the considered problem to a system of nonlinear algebraic equations. Some examples are presented to illustrate the efficiency and accuracy of the method.

  12. Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method

    Molabahrami, A. [Department of Mathematics, Ilam University, PO Box 69315516, Ilam (Iran, Islamic Republic of)], E-mail:; Khani, F. [Department of Mathematics, Ilam University, PO Box 69315516, Ilam (Iran, Islamic Republic of); Bakhtar Institute of Higher Education, PO Box 696, Ilam (Iran, Islamic Republic of)], E-mail:; Hamedi-Nezhad, S. [Bakhtar Institute of Higher Education, PO Box 696, Ilam (Iran, Islamic Republic of)


    In this Letter, the He's homotopy perturbation method (HPM) to finding the soliton solutions of the two-dimensional Korteweg-de Vries Burgers' equation (tdKdVB) for the initial conditions was applied. Numerical solutions of the equation were obtained. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. The results reveal that the HPM is very effective and simple.

  13. Quadrature Rules and Iterative Method for Numerical Solution of Two-Dimensional Fuzzy Integral Equations

    S. M. Sadatrasoul


    Full Text Available We introduce some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy-number-valued functions. We also give error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, we propose an iterative procedure based on quadrature formula to solve two-dimensional linear fuzzy Fredholm integral equations of the second kind (2DFFLIE2, and we present the error estimation of the proposed method. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

  14. The tyger phenomenon for the Galerkin-truncated Burgers and Euler equations

    Ray, Samriddhi Sankar; Nazarenko, Sergei; Matsumoto, Takeshi


    It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wavenumbers in excess of a threshold $\\kg$ exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. At large $\\kg$, for smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation which we call a "tyger", is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features (shocks, layers with strong vorticity gradients, etc). These tygers appear when complex-space singularities come within one Galerkin wavelength $\\lambdag = 2\\pi/\\kg$ from the real domain and typically arise far away from preexisting small-scale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first - in the Burgers case at the time of appearance of the first shock their ...

  15. Singularity formation for one dimensional full Euler equations

    Pan, Ronghua; Zhu, Yi


    We investigate the basic open question on the global existence v.s. finite time blow-up phenomena of classical solutions for the one-dimensional compressible Euler equations of adiabatic flow. For isentropic flows, it is well-known that the solutions develop singularity if and only if initial data contain any compression (the Riemann variables have negative spatial derivative). The situation for non-isentropic flow is not quite clear so far, due to the presence of non-constant entropy. In [4], it is shown that initial weak compressions do not necessarily develop singularity in finite time, unless the compression is strong enough for general data. In this paper, we identify a class of solutions of the full (non-isentropic) Euler equations, developing singularity in finite time even though their initial data do not contain any compression. This is in sharp contrast to the isentropic flow.


    Nicolae Cîndea


    Full Text Available In this work we prove that the exact internal observability for theEuler-Bernoulli equation is robust with respect to a class of linear perturbations. Our results yield,in particular,that for rectangular domains we have the exact observability in an arbitrarily small time and with an arbitrarily small observation region. The usual method of tackling lower order terms,using Carleman estimates, cannot be applied in this context. More precisely, it is not known if Carleman estimates hold for the evolution Euler-Bernoulli equation with arbitrarily small observation region. Therefore we use a method combining frequency domain techniques,a compactness-uniqueness argument and a Carleman estimate for elliptic problems.

  17. Existence of Weak Solutions for the Incompressible Euler Equations

    Wiedemann, Emil


    Using a recent result of C. De Lellis and L. Sz\\'{e}kelyhidi Jr. we show that, in the case of periodic boundary conditions and for dimension greater or equal 2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data $v_0$, where $v_0$ may be any solenoidal $L^2$-vectorfield. In addition, the energy of these solutions is bounded in time.

  18. Euler equation existence, non-uniqueness and mesh converged statistics.

    Glimm, James; Sharp, David H; Lim, Hyunkyung; Kaufman, Ryan; Hu, Wenlin


    We review existence and non-uniqueness results for the Euler equation of fluid flow. These results are placed in the context of physical models and their solutions. Non-uniqueness is in direct conflict with the purpose of practical simulations, so that a mitigating strategy, outlined here, is important. We illustrate these issues in an examination of mesh converged turbulent statistics, with comparison to laboratory experiments.

  19. Viscous Regularization of the Euler Equations and Entropy Principles

    Guermond, Jean-Luc


    This paper investigates a general class of viscous regularizations of the compressible Euler equations. A unique regularization is identified that is compatible with all the generalized entropies, à la [Harten et al., SIAM J. Numer. Anal., 35 (1998), pp. 2117-2127], and satisfies the minimum entropy principle. A connection with a recently proposed phenomenological model by [H. Brenner, Phys. A, 370 (2006), pp. 190-224] is made. © 2014 Society for Industrial and Applied Mathematics.

  20. Blowup phenomena for the compressible euler and euler-poisson equations with initial functional conditions.

    Wong, Sen; Yuen, Manwai


    We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in R (N) . For time t ≥ 0, we can define a functional H(t) associated with the solution of the equations and some testing function f. When the pressure function P of the governing equations is of the form P = Kρ (γ) , where ρ is the density function, K is a constant, and γ > 1, we can show that the nontrivial C (1) solutions with nonslip boundary condition will blow up in finite time if H(0) satisfies some initial functional conditions defined by the integrals of f. Examples of the testing functions include r (N-1)ln(r + 1), r (N-1) e (r) , r (N-1)(r (3) - 3r (2) + 3r + ε), r (N-1)sin((π/2)(r/R)), and r (N-1)sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.

  1. The periodic b-equation and Euler equations on the circle

    Escher, J


    In this note we show that the periodic b-equation can only be realized as an Euler equation on the Lie group Diff(S^1) of all smooth and orientiation preserving diffeomorphisms on the cirlce if b=2, i.e. for the Camassa-Holm equation. In this case the inertia operator generating the metric on Diff(S^1) is given by A=1-d^2/dx^2. In contrast, the Degasperis-Procesi equation, for which b=3, is not an Euler equation on Diff(S^1) for any inertia operator. Our result generalizes a recent result of B. Kolev.

  2. The Analysis of the Two-dimensional Diffusion Equation With a Source

    Sunday Augustus REJU


    Full Text Available This study presents a new variant analysis and simulations of the two-dimensional energized wave equation remarkably different from the diffusion equations studied earlier studied. The objective functional and the dynamical energized wave are penalized to form a function called the Hamiltonian function. From this function, we obtained the necessary conditions for the optimal solutions using the maximum principle. By applying the Fourier solution to the first order differential equation, the analytical solutions for the state and control are obtained. The solutions are simulated to give visual physical interpretation of the waves and the numerical values.

  3. Stellar fibril magnetic systems. II - Two-dimensional magnetohydrodynamic equations. III - Convective counterflow

    Parker, E. N.


    The dynamics of magnetic fibrils in the convective zone of a star is investigated analytically, deriving mean-field equations for the two-dimensional transverse motion of an incompressible fluid containing numerous small widely spaced circular cylinders. The equations of Parker (1982) are extended to account for the inertial effects of local flow around the cylinders. The linear field equation for the stream function at the onset of convection is then rewritten, neglecting large-scale heat transport, and used to construct a model of convective counterflow. The Kelvin impulse and fluid momentum, convective motion initiated by a horizontal impulse, and the effects of a viscous boundary layer are considered in appendices.

  4. An implicit logarithmic finite-difference technique for two dimensional coupled viscous Burgers’ equation

    Srivastava, Vineet K., E-mail: [ISRO Telemetry, Tracking and Command Network (ISTRAC), Bangalore-560058 (India); Awasthi, Mukesh K. [Department of Mathematics, University of Petroleum and Energy Studies, Dehradun-248007 (India); Singh, Sarita [Department of Mathematics, WIT- Uttarakhand Technical University, Dehradun-248007 (India)


    This article describes a new implicit finite-difference method: an implicit logarithmic finite-difference method (I-LFDM), for the numerical solution of two dimensional time-dependent coupled viscous Burgers’ equation on the uniform grid points. As the Burgers’ equation is nonlinear, the proposed technique leads to a system of nonlinear systems, which is solved by Newton's iterative method at each time step. Computed solutions are compared with the analytical solutions and those already available in the literature and it is clearly shown that the results obtained using the method is precise and reliable for solving Burgers’ equation.

  5. An implicit logarithmic finite-difference technique for two dimensional coupled viscous Burgers’ equation

    Vineet K. Srivastava


    Full Text Available This article describes a new implicit finite-difference method: an implicit logarithmic finite-difference method (I-LFDM, for the numerical solution of two dimensional time-dependent coupled viscous Burgers’ equation on the uniform grid points. As the Burgers’ equation is nonlinear, the proposed technique leads to a system of nonlinear systems, which is solved by Newton's iterative method at each time step. Computed solutions are compared with the analytical solutions and those already available in the literature and it is clearly shown that the results obtained using the method is precise and reliable for solving Burgers’ equation.

  6. Multi-Symplectic Splitting Method for Two-Dimensional Nonlinear Schriidinger Equation

    陈亚铭; 朱华君; 宋松和


    Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multisymplectic splitting (MSS) method to solve the two-dimensional nonlinear Schrodinger equation (2D-NLSE) in this paper. It is further shown that the method constructed in this way preserve the global symplectieity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness of the proposed method.

  7. Third order finite volume evolution Galerkin (FVEG) methods for two-dimensional wave equation system

    Lukácová-Medvid'ová, Maria; Warnecke, Gerald; Zahaykah, Yousef


    The subject of the paper is the derivation and analysis of third order finite volume evolution Galerkin schemes for the two-dimensional wave equation system. To achieve this the first order approximate evolution operator is considered. A recovery stage is carried out at each level to generate a piecewise polynomial approximation from the piecewise constants, to feed into the calculation of the fluxes. We estimate the truncation error and give numerical examples to demonstrate the higher order...

  8. Measurement of the Equation of State of the Two-Dimensional Hubbard Model

    Miller, Luke; Cocchi, Eugenio; Drewes, Jan; Koschorreck, Marco; Pertot, Daniel; Brennecke, Ferdinand; Koehl, Michael


    The subtle interplay between kinetic energy, interactions and dimensionality challenges our comprehension of strongly-correlated physics observed, for example, in the solid state. In this quest, the Hubbard model has emerged as a conceptually simple, yet rich model describing such physics. Here we present an experimental determination of the equation of state of the repulsive two-dimensional Hubbard model over a broad range of interactions, 0 constitute benchmarks for state-of-the-art theoretical approaches.

  9. Quadrature-free spline method for two-dimensional Navier-Stokes equation

    HU Xian-liang; HAN Dan-fu


    In this paper,a quadrature-free scheme of spline method for two-dimensional Navier-Stokes equation is derived,which can dramatically improve the efficiency of spline method for fluid problems proposed by Lai and Wenston(2004). Additionally,the explicit formulation for boundary condition with up to second order derivatives is presented. The numerical simulations on several benchmark problems show that the scheme is very efficient.

  10. Symmetries of the Euler compressible flow equations for general equation of state

    Boyd, Zachary M. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Ramsey, Scott D. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Baty, Roy S. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)


    The Euler compressible flow equations exhibit different Lie symmetries depending on the equation of state (EOS) of the medium in which the flow occurs. This means that, in general, different types of similarity solution will be available in different flow media. We present a comprehensive classification of all EOS’s to which the Euler equations apply, based on the Lie symmetries admitted by the corresponding flow equations, restricting to the case of 1-D planar, cylindrical, or spherical geometry. The results are conveniently summarized in tables. This analysis also clarifies past work by Axford and Ovsiannikov on symmetry classification.

  11. Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation

    Jiu, Quansen; Wu, Jiahong; Yang, Wanrong


    This paper studies solutions of the two-dimensional incompressible Boussinesq equations with fractional dissipation. The spatial domain is a periodic box. The Boussinesq equations concerned here govern the coupled evolution of the fluid velocity and the temperature and have applications in fluid mechanics and geophysics. When the dissipation is in the supercritical regime (the sum of the fractional powers of the Laplacians in the velocity and the temperature equations is less than 1), the classical solutions of the Boussinesq equations are not known to be global in time. Leray-Hopf type weak solutions do exist. This paper proves that such weak solutions become eventually regular (smooth after some time ) when the fractional Laplacian powers are in a suitable supercritical range. This eventual regularity is established by exploiting the regularity of a combined quantity of the vorticity and the temperature as well as the eventual regularity of a generalized supercritical surface quasi-geostrophic equation.

  12. GPU Accelerated Spectral Element Methods: 3D Euler equations

    Abdi, D. S.; Wilcox, L.; Giraldo, F.; Warburton, T.


    A GPU accelerated nodal discontinuous Galerkin method for the solution of three dimensional Euler equations is presented. The Euler equations are nonlinear hyperbolic equations that are widely used in Numerical Weather Prediction (NWP). Therefore, acceleration of the method plays an important practical role in not only getting daily forecasts faster but also in obtaining more accurate (high resolution) results. The equation sets used in our atomospheric model NUMA (non-hydrostatic unified model of the atmosphere) take into consideration non-hydrostatic effects that become more important with high resolution. We use algorithms suitable for the single instruction multiple thread (SIMT) architecture of GPUs to accelerate solution by an order of magnitude (20x) relative to CPU implementation. For portability to heterogeneous computing environment, we use a new programming language OCCA, which can be cross-compiled to either OpenCL, CUDA or OpenMP at runtime. Finally, the accuracy and performance of our GPU implementations are veried using several benchmark problems representative of different scales of atmospheric dynamics.

  13. Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

    Kouatchou, Jules


    In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.

  14. Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations

    Ablowitz, Mark J.; Demirci, Ali; Ma, Yi-Ping


    Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2 + 1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.

  15. Exploration of POD-Galerkin Techniques for Developing Reduced Order Models of the Euler Equations


    Models of the Euler Equations 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Mundis, N., Edoh, A. and Sankaran...for describing combustion response to specific excitations using Euler equations as a continued work from a previous studies using a reaction...eigen-bases. For purposes of this study, a linearized version of the Euler equations is employed. The knowledge obtained from previous scalar equation

  16. Anti-periodic traveling wave solution to a forced two-dimensional generalized KdV-Burgers equation

    TAN Junyu


    The anti-periodic traveling wave solutions to a forced two-dimensional generalized KdV-Burgers equation are studied.Some theorems concerning the boundness, existence and uniqueness of the solution to this equation are proved.

  17. Analytical solutions of the two-dimensional Dirac equation for a topological channel intersection

    Anglin, J. R.; Schulz, A.


    Numerical simulations in a tight-binding model have shown that an intersection of topologically protected one-dimensional chiral channels can function as a beam splitter for noninteracting fermions on a two-dimensional lattice [Qiao, Jung, and MacDonald, Nano Lett. 11, 3453 (2011), 10.1021/nl201941f; Qiao et al., Phys. Rev. Lett. 112, 206601 (2014), 10.1103/PhysRevLett.112.206601]. Here we confirm this result analytically in the corresponding continuum k .p model, by solving the associated two-dimensional Dirac equation, in the presence of a "checkerboard" potential that provides a right-angled intersection between two zero-line modes. The method by which we obtain our analytical solutions is systematic and potentially generalizable to similar problems involving intersections of one-dimensional systems.

  18. Protein Conformational Change Based on a Two-dimensional Generalized Langevin Equation

    Ying-xi Wang; Shuang-mu Linguang; Nan-rong Zhao; Yi-jing Yan


    A two-dimensional generalized Langevin equation is proposed to describe the protein conformational change,compatible to the electron transfer process governed by atomic packing density model.We assume a fractional Gaussian noise and a white noise through bond and through space coordinates respectively,and introduce the coupling effect coming from both fluctuations and equilibrium variances.The general expressions for autocorrelation functions of distance fluctuation and fluorescence lifetime variation are derived,based on which the exact conformational change dynamics can be evaluated with the aid of numerical Laplace inversion technique.We explicitly elaborate the short time and long time approximations.The relationship between the two-dimensional description and the one-dimensional theory is also discussed.

  19. Optimum Transonic Airfoils Based on the Euler Equations

    Iollo, Angelo; Salas, Manuel, D.


    We solve the problem of determining airfoils that approximate, in a least square sense, given surface pressure distributions in transonic flight regimes. The flow is modeled by means of the Euler equations and the solution procedure is an adjoint- based minimization algorithm that makes use of the inverse Theodorsen transform in order to parameterize the airfoil. Fast convergence to the optimal solution is obtained by means of the pseudo-time method. Results are obtained using three different pressure distributions for several free stream conditions. The airfoils obtained have given a trailing edge angle.

  20. NURBS-enhanced finite element method for Euler equations

    Sevilla Cárdenas, Rubén; Fernandez Mendez, Sonia; Huerta, Antonio , coaut.


    This is the pre-peer reviewed version of the following article: Sevilla, R.; Fernandez, S.; Huerta, A. NURBS-enhanced finite element method for Euler equations. "International journal for numerical methods in fluids", Juliol 2008, vol. 57, núm. 9, p. 1051-1069., which has been published in final form at In this work, the NURBS-enhanced finite element method (NEFEM) is combined with a discontinuous Galerkin (DG) formulation for t...

  1. A novel numerical flux for the 3D Euler equations with general equation of state

    Toro, Eleuterio F.


    Here we extend the flux vector splitting approach recently proposed in (E F Toro and M E Vázquez-Cendón. Flux splitting schemes for the Euler equations. Computers and Fluids. Vol. 70, Pages 1-12, 2012). The scheme was originally presented for the 1D Euler equations for ideal gases and its extension presented in this paper is threefold: (i) we solve the three-dimensional Euler equations on general meshes; (ii) we use a general equation of state; and (iii) we achieve high order of accuracy in both space and time through application of the semi-discrete ADER methodology on general meshes. The resulting methods are systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems. Formal high accuracy is assessed through convergence rates studies for schemes of up to 4th order of accuracy in both space and time on unstructured meshes.

  2. Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

    Chavanis, Pierre-Henri


    We complete the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and H\\"uckel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-H\\"uckel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-si...

  3. Potentially Singular Solutions of the 3D Incompressible Euler Equations

    Luo, Guo


    Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \\times 10^{12})^{2}$ near the point of the singularity, we are able to advance the solution up to $\\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \\approx 0.0035056$, while achieving a \\emph{pointw...

  4. Equation for self-similar singularity of Euler 3D

    Pomeau, Yves


    The equations for a self-similar solution of an inviscid incompressible fluid are mapped into an integral equation which hopefully can be solved by iteration. It is argued that the exponent of the similarity are ruled by Kelvin's theorem of conservation of circulation. The end result is an iteration with a nonlinear term entering in a kernel given by a 3D integral (in general 3D flow) or 2D (for swirling flows), which seems to be within reach of present day computational power. Because of the slow decay of the similarity solution at large distances the total energy is diverging and recent mathematical results excluding a solution of the self-similar solution of Euler equation do not apply.

  5. Perturbations of Half-Linear Euler Differential Equation and Transformations of Modified Riccati Equation

    Ondřej Došlý


    Full Text Available We investigate transformations of the modified Riccati differential equation and the obtained results we apply in the investigation of oscillatory properties of perturbed half-linear Euler differential equation. A perturbation is also allowed in the differential term.

  6. Fourier solution of two-dimensional Navier Stokes equation with periodic boundary conditions and incompressible flow

    Kuiper, Logan K


    An approximate solution to the two dimensional Navier Stokes equation with periodic boundary conditions is obtained by representing the x any y components of fluid velocity with complex Fourier basis vectors. The chosen space of basis vectors is finite to allow for numerical calculations, but of variable size. Comparisons of the resulting approximate solutions as they vary with the size of the chosen vector space allow for extrapolation to an infinite basis vector space. Results suggest that such a solution, with the full basis vector space and which would give the exact solution, would fail for certain initial velocity configurations when initial velocity and time t exceed certain limits.

  7. On two-dimensional large-scale primitive equations in oceanic dynamics(Ⅰ)

    HUANG Dai-wen; GUO Bo-ling


    The initial boundary value problem for the two-dimensional primitive equations of large scale oceanic motion in geophysics is considered.It is assumed that the depth of the ocean is a positive constant.Firstly,if the initial data are square integrable,then by Fadeo-Galerkin method,the existence of the global weak solutions for the problem is obtained.Secondly, if the initial data and their vertical derivatives axe all square integrable,then by Faedo-Galerkin method and anisotropic inequalities,the existerce and uniqueness of the giobal weakly strong solution for the above initial boundary problem axe obtained.

  8. On two-dimensional large-scale primitive equations in oceanic dynamics(Ⅱ)

    HUANG Dai-wen; GUO Bo-ling


    The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially.Here the depth of the ocean is positive but not always a constant.By Faedo-Galerkin method and anisotropic inequalities,the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained.Moreover,by studying the asymptotic behavior of solutions for the above problem,the energy is exponential decay with time is proved.

  9. Cellular neural network analysis for two-dimensional bioheat transfer equation.

    Niu, J H; Wang, H Z; Zhang, H X; Yan, J Y; Zhu, Y S


    The cellular neural network (CNN) method is applied to solve the Pennes bioheat transfer equation, and its feasibility is demonstrated. Numerical solutions were obtained for a cellular neural network for a two-dimensional steady-state temperature field obtained from focused and unfocused ultrasound heat sources. Transient-state temperature fields were also studied and compared with experimental results obtained elsewhere. The cellular neural networks' key features of asynchronous parallel processing, continuous-time dynamics and local interaction enable real-time temperature field estimation for clinical hyperthermia.

  10. Ultrashort light bullets described by the two-dimensional sine-Gordon equation

    Leblond, Hervé; 10.1103/PHYSREVA.81.063815


    By using a reductive perturbation technique applied to a two-level model, this study puts forward a generic two-dimensional sine-Gordon evolution equation governing the propagation of femtosecond spatiotemporal optical solitons in Kerr media beyond the slowly varying envelope approximation. Direct numerical simulations show that, in contrast to the long-wave approximation, no collapse occurs, and that robust (2+1)-dimensional ultrashort light bullets may form from adequately chosen few-cycle input spatiotemporal wave forms. In contrast to the case of quadratic nonlinearity, the light bullets oscillate in both space and time and are therefore not steady-state lumps.

  11. Potentially singular solutions of the 3D axisymmetric Euler equations.

    Luo, Guo; Hou, Thomas Y


    The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(ts - t)(-2.46), where ts ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ(2) = 0.003505 at which time the vorticity amplifies by more than (3 × 10(8))-fold and the maximum mesh resolution exceeds (3 × 10(12))(2). The vorticity vector is observed to maintain four significant digits throughout the computations.

  12. Iterative methods for compressible Navier-Stokes and Euler equations

    Tang, W.P.; Forsyth, P.A.


    This workshop will focus on methods for solution of compressible Navier-Stokes and Euler equations. In particular, attention will be focused on the interaction between the methods used to solve the non-linear algebraic equations (e.g. full Newton or first order Jacobian) and the resulting large sparse systems. Various types of block and incomplete LU factorization will be discussed, as well as stability issues, and the use of Newton-Krylov methods. These techniques will be demonstrated on a variety of model transonic and supersonic airfoil problems. Applications to industrial CFD problems will also be presented. Experience with the use of C++ for solution of large scale problems will also be discussed. The format for this workshop will be four fifteen minute talks, followed by a roundtable discussion.

  13. Dynamics in discrete two-dimensional nonlinear Schrödinger equations in the presence of point defects

    Christiansen, Peter Leth; Gaididei, Yuri Borisovich; Rasmussen, Kim


    The dynamics of two-dimensional discrete structures is studied in the framework of the generalized two-dimensional discrete nonlinear Schrodinger equation. The nonlinear coupling in the form of the Ablowitz-Ladik nonlinearity and point impurities is taken into account. The stability properties...

  14. Exact Solutions of Two-dimensional and Tri-dimensional Consolidation Equations

    Di Francesco, Romolo


    The exact solution of Terzaghi's consolidation equation has further highlighted the limits of this theory in the one-dimensional field as, like Taylor's approximate solution, it overestimates the decay times of the phenomenon; on the other hand, one only needs to think about the accumulation pattern of sedimentary-basin soils to understand how their internal structure fits in more with the model of transversely isotropic medium, so as to result in the development of two- and three-dimensional consolidation models. This is the reason why, using Terzaghi's theory and his exact solution as starting point, two-dimensional and three-dimensional consolidation equations have been proposed, in an attempt to find their corresponding exact solutions which constitute more reliable forecasting models. Lastly, results show how this phenomenon is predominantly influenced by the dimensions of the horizontal plane affected by soil consolidation and permeabilities that behave according to three coordinate axes.

  15. Generalized scale-invariant solutions to the two-dimensional stationary Navier-Stokes equations

    Guillod, Julien


    New explicit solutions to the incompressible Navier-Stokes equations in $\\mathbb{R}^{2}\\setminus\\left\\{ \\boldsymbol{0}\\right\\}$ are determined, which generalize the scale-invariant solutions found by Hamel. These new solutions are invariant under a particular combination of the scaling and rotational symmetries. They are the only solutions invariant under this new symmetry in the same way as the Hamel solutions are the only scale-invariant solutions. While the Hamel solutions are parameterized by a discrete parameter $n$, the flux $\\Phi$ and an angle $\\theta_{0}$, the new solutions generalize the Hamel solutions by introducing an additional parameter $a$ which produces a rotation. The new solutions decay like $\\left|\\boldsymbol{x}\\right|^{-1}$ as the Hamel solutions, and exhibit spiral behavior. The new variety of asymptotes induced by the existence of these solutions further emphasizes the difficulties faced when trying to establish the asymptotic behavior of the Navier-Stokes equations in a two-dimensional ...

  16. Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions

    R. Naz


    Full Text Available We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass density g(x, and the applied load denoted by f(u, a function of transverse displacement u(t,x. The complete Lie group classification is obtained for different forms of the variable lineal mass density g(x and applied load f(u. The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of g(x. For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature when g(x is constant with variable applied load f(u. For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.

  17. Noether Symmetry Analysis of the Dynamic Euler-Bernoulli Beam Equation

    Johnpillai, A. G.; Mahomed, K. S.; Harley, C.; Mahomed, F. M.


    We study the fourth-order dynamic Euler-Bernoulli beam equation from the Noether symmetry viewpoint. This was earlier considered for the Lie symmetry classification. We obtain the Noether symmetry classification of the equation with respect to the applied load, which is a function of the dependent variable of the underlying equation. We find that the principal Noether symmetry algebra is two-dimensional when the load function is arbitrary and extends for linear and power law cases. For all cases, for each of the Noether symmetries associated with the usual Lagrangian, we construct conservation laws for the equation via the Noether theorem. We also provide a basis of conservation laws by using the adjoint algebra. The Noether symmetries pick out the special value of the power law, which is -7. We consider the Noether symmetry reduction for this special case, which gives rise to a first integral that is used for our numerical code. For this, we then find numerical solutions using an in-built function in MATLAB called bvp4c, which is a boundary value solver for differential equations that are depicted in five figures. The physical solutions obtained are for the deflection of the beam with an increase in displacement. These are given in four figures and discussed.


    S Pamuk; N Pamuk


      In this paper, we obtain the particular exact solutions of the two-dimensional heat and mass transfer equation with power-law temperature-dependent thermal con- ductivity using the Adomian's decomposition method...

  19. Non-Lie Symmetry Group and New Exact Solutions for the Two-Dimensional KdV-Burgers Equation

    WANG Hong; TIAN Ying-Hui; CHEN Han-Lin


    @@ By using the modified Clarkson-Kruskal (CK) direct method, we obtain the non-Lie symmetry group of the two-dimensional KdV-Burgers equation.Under some constraint conditions, Lie point symmetry is also obtained.Through the symmetry group, some new exact solutions of the two-dimensional KdV-Burgers equation are found.%By using the modified Clarkson-Kruskal (CK) direct method, we obtain the non-Lie symmetry group of the two-dimensional KdV-Burgers equation. Under some constraint conditions, Lie point symmetry is also obtained.Through the symmetry group, some new exact solutions of the two-dimensional KdV-Burgers equation are found.

  20. Generalized Burgers equations and Euler-Painlevé transcendents. I

    Sachdev, P. L.; Nair, K. R. C.; Tikekar, V. G.


    Initial-value problems for the generalized Burgers equation (GBE) ut+u βux+λuα =(δ/2)uxx are discussed for the single hump type of initial data—both continuous and discontinuous. The numerical solution is carried to the self-similar ``intermediate asymptotic'' regime when the solution is given analytically by the self-similar form. The nonlinear (transformed) ordinary differential equations (ODE's) describing the self-similar form are generalizations of a class discussed by Euler and Painlevé and quoted by Kamke. These ODE's are new, and it is postulated that they characterize GBE's in the same manner as the Painlevé equations categorize the Kortweg-de Vries (KdV) type. A connection problem for some related ODE's satisfying proper asymptotic conditions at x=±∞, is solved. The range of amplitude parameter is found for which the solution of the connection problem exists. The other solutions of the above GBE, which display several interesting features such as peaking, breaking, and a long shelf on the left for negative values of the damping coefficient λ, are also discussed. The results are compared with those holding for the modified KdV equation with damping.

  1. Regularized inversion of a two-dimensional integral equation with applications in borehole induction measurements

    Arikan, Orhan


    Well bore measurements of conductivity, gravity, and surface measurements of magnetotelluric fields can be modeled as a two-dimensional integral equation with additive measurement noise. The governing integral equation has the form of convolution in the first dimension and projection in the second dimension. However, these two operations are not in separable form. In these applications, given a set of measurements, efficient and robust estimation of the underlying physical property is required. For this purpose, a regularized inversion algorithm for the governing integral equation is presented in this paper. Singular value decomposition of the measurement kernels is used to exploit convolution-projection structure of the integral equation, leading to a form where measurements are related to the physical property by a two-stage operation: projection followed by convolution. On the other hand, estimation of the physical property can be carried out by a two-stage inversion algorithm: deconvolution followed by back projection. A regularization method for the required multichannel deconvolution is given. Some important details of the algorithm are addressed in an application to wellbore induction measurements of conductivity.

  2. Convergence of the Vlasov-Poisson-Fokker- Planck system to the incompressible Euler equations


    We establish the convergence of the Vlasov-Poisson-Fokker-Planck system to the incompressible Euler equations in this paper. The convergence is rigorously proved on the time interval where the smooth solution to the incompressible Euler equations exists. The proof relies on the compactness argument and the so-called relative-entropy method.

  3. Euler-Lagrange Equations of Networks with Higher-Order Elements

    Z. Biolek


    Full Text Available The paper suggests a generalization of the classic Euler-Lagrange equation for circuits compounded of arbitrary elements from Chua’s periodic table. Newly defined potential functions for general (α, β elements are used for the construction of generalized Lagrangians and generalized dissipative functions. Also procedures of drawing the Euler-Lagrange equations are demonstrated.

  4. Ill-Posedness of the Hydrostatic Euler and Singular Vlasov Equations

    Han-Kwan, Daniel; Nguyen, Toan T.


    In this paper, we develop an abstract framework to establish ill-posedness, in the sense of Hadamard, for some nonlocal PDEs displaying unbounded unstable spectra. We apply this to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system.

  5. Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

    Kazakov, A. Ya.; Slavyanov, S. Yu.


    Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation.

  6. Extrapolation of Nystrom solution for two dimensional nonlinear Fredholm integral equations

    Guoqiang, Han; Jiong, Wang


    In this paper, we analyze the existence of asymptotic error expansion of the Nystrom solution for two-dimensional nonlinear Fredholm integral equations of the second kind. We show that the Nystrom solution admits an error expansion in powers of the step-size h and the step-size k. For a special choice of the numerical quadrature, the leading terms in the error expansion for the Nystrom solution contain only even powers of h and k, beginning with terms h2p and k2q. These expansions are useful for the application of Richardson extrapolation and for obtaining sharper error bounds. Numerical examples show that how Richardson extrapolation gives a remarkable increase of precision, in addition to faster convergence.

  7. The solution of the two-dimensional sine-Gordon equation using the method of lines

    Bratsos, A. G.


    The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor-corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.

  8. Numerical approach for solving kinetic equations in two-dimensional case on hybrid computational clusters

    Malkov, Ewgenij A.; Poleshkin, Sergey O.; Kudryavtsev, Alexey N.; Shershnev, Anton A.


    The paper presents the software implementation of the Boltzmann equation solver based on the deterministic finite-difference method. The solver allows one to carry out parallel computations of rarefied flows on a hybrid computational cluster with arbitrary number of central processor units (CPU) and graphical processor units (GPU). Employment of GPUs leads to a significant acceleration of the computations, which enables us to simulate two-dimensional flows with high resolution in a reasonable time. The developed numerical code was validated by comparing the obtained solutions with the Direct Simulation Monte Carlo (DSMC) data. For this purpose the supersonic flow past a flat plate at zero angle of attack is used as a test case.

  9. Universal equations of unsteady two-dimensional MHD boundary layer whose temperature varies with time

    Boričić Zoran


    Full Text Available This paper concerns with unsteady two-dimensional temperature laminar magnetohydrodynamic (MHD boundary layer of incompressible fluid. It is assumed that induction of outer magnetic field is function of longitudinal coordinate with force lines perpendicular to the body surface on which boundary layer forms. Outer electric filed is neglected and magnetic Reynolds number is significantly lower then one i.e. considered problem is in inductionless approximation. Characteristic properties of fluid are constant because velocity of flow is much lower than speed of light and temperature difference is small enough (under 50ºC . Introduced assumptions simplify considered problem in sake of mathematical solving, but adopted physical model is interesting from practical point of view, because its relation with large number of technically significant MHD flows. Obtained partial differential equations can be solved with modern numerical methods for every particular problem. Conclusions based on these solutions are related only with specific temperature MHD boundary layer problem. In this paper, quite different approach is used. First new variables are introduced and then sets of similarity parameters which transform equations on the form which don't contain inside and in corresponding boundary conditions characteristics of particular problems and in that sense equations are considered as universal. Obtained universal equations in appropriate approximation can be solved numerically once for all. So-called universal solutions of equations can be used to carry out general conclusions about temperature MHD boundary layer and for calculation of arbitrary particular problems. To calculate any particular problem it is necessary also to solve corresponding momentum integral equation.

  10. Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation

    Lannes, David


    A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet-Neumann operator and we show that all the asymptotic models can be recovered by a simple asymptotic expansion of this operator, in function of the shallowness parameter (shallow water limit) or the steepness parameter (deep water limit). Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to uneven bottom the approach developed by Matsuno \\cite{matsuno3} and Choi \\cite{choi}. This model is still valid in shallow water but with less precision than what can be achieved with Green-Naghdi model, when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equati...

  11. Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

    GONG Lun-Xun; CAO Jian-Li; PAN Jun-Ting; ZHANG Hua; JIAO Wan-Tang


    Based on the second integrable case of known two-dimensional Hamiltonian system with a quartic potential, we propose a 4×4 matrix spectral problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differential equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable case of the two-dimensional Hamiltonian system.

  12. Solitary wave solutions of two-dimensional nonlinear Kadomtsev–Petviashvili dynamic equation in dust-acoustic plasmas



    Nonlinear two-dimensional Kadomtsev–Petviashvili (KP) equation governs the behaviour of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions. By using the reductive perturbation method, the two-dimensional dust-acoustic solitary waves (DASWs) in unmagnetized cold plasma consisting of dust fluid, ions and electrons lead to a KP equation. We derived the solitary travelling wave solutions of the twodimensional nonlinear KP equation by implementing sech–tanh, sinh–cosh, extended direct algebraic and fraction direct algebraicmethods. We found the electrostatic field potential and electric field in the form travellingwave solutions for two-dimensional nonlinear KP equation. The solutions for the KP equation obtained by using these methods can be demonstrated precisely and efficiency. As an illustration, we used the readymade package of $\\it{Mathematica}$ program 10.1 to solve the original problem. These solutions are in good agreement with the analytical one.

  13. Delta Shocks and Vacuum States in Vanishing Pressure Limits of Solutions to the Relativistic Euler Equations

    Gan YIN; Wancheng SHENG


    The Riemann problems for the Euler system of conservation laws of energy and momentum in special relativity as pressure vanishes are considered. The Riemann solutions for the pressureless relativistic Euler equations are obtained constructively. There are two kinds of solutions, the one involves delta shock wave and the other involves vacuum. The authors prove that these two kinds of solutions are the limits of the solutions as pressure vanishes in the Euler system of conservation laws of energy and momentum in special relativity.

  14. A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method

    Chunye Gong


    Full Text Available It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE with iterative implicit finite difference method is O(MxMyN2. In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16–4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.

  15. 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.

  16. The First-Order Euler-Lagrange equations and some of their uses

    Adam, C


    In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.

  17. The first-order Euler-Lagrange equations and some of their uses

    Adam, C.; Santamaria, F. [Departamento de Física de Partículas and Instituto Galego de Física de Altas Enerxias (IGFAE),Campus Vida, E-15782 Santiago de Compostela (Spain)


    In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.

  18. Numerical Investigations on Hybrid Fuzzy Fractional Differential Equations by Improved Fractional Euler Method

    D. Vivek


    Full Text Available In this paper, the improved Euler method is used for solving hybrid fuzzy fractional differential equations (HFFDE of order $q \\in (0, 1 $ under Caputo-type fuzzy fractional derivatives. This method is based on the fractional Euler method and generalized Taylor's formula. The accuracy and efficiency of the proposed method is demonstrated by solving numerical examples.

  19. Computational Aerodynamics Based on the Euler Equations (L’aerodynamique Numerique a Partir des Equations d’Euler)


    dquations d’Euler) Edited by * S J.W. SLOOFF National Aerospace Laboratory NLR Anthony Fokkerweg 2 1059 CM Amsterdam Netherlands Dr. W. SCHMIDT Air Vehicle...CHAKRAVARTHY. S. R., RIBA , W.* * T.. BYERLY. J. and DRESSER. H. S.. "Multi-Zone Euler 41. DESLANDES R.M.. "Theoretisebe bestimmung der Marching Technique

  20. Quenched dynamics of two-dimensional solitons and vortices in the Gross-Pitaevskii equation

    Chen, Qian-Yong; Malomed, Boris A


    We consider a two-dimensional (2D) counterpart of the experiment that led to the creation of quasi-1D bright solitons in Bose-Einstein condensates (BECs) [Nature 417, 150--153 (2002)]. We start by identifying the ground state of the 2D Gross-Pitaevskii equation for repulsive interactions, with a harmonic-oscillator (HO) trap, and with or without an optical lattice (OL). Subsequently, we switch the sign of the interaction to induce interatomic attraction and monitor the ensuing dynamics. Regions of the stable self-trapping and catastrophic collapse of 2D fundamental solitons are identified in the parameter plane of the OL strength and BEC norm. The increase of the OL strength expands the persistence domain for the solitons to larger norms. For single-charged solitary vortices, in addition to the survival and collapse regimes, an intermediate one is identified, where the vortex resists the collapse but loses its structure, transforming into a fundamental soliton. The same setting may also be implemented in the ...

  1. General solution of the Dirac equation for quasi-two-dimensional electrons

    Eremko, Alexander, E-mail: [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); Brizhik, Larissa, E-mail: [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); Loktev, Vadim, E-mail: [Bogolyubov Institute for Theoretical Physics, Metrologichna Str., 14-b, Kyiv, 03680 (Ukraine); National Technical University of Ukraine “KPI”, Peremohy av., 37, Kyiv, 03056 (Ukraine)


    The general solution of the Dirac equation for quasi-two-dimensional electrons confined in an asymmetric quantum well, is found. The energy spectrum of such a system is exactly calculated using special unitary operator and is shown to depend on the electron spin polarization. This solution contains free parameters, whose variation continuously transforms one known particular solution into another. As an example, two different cases are considered in detail: electron in a deep and in a strongly asymmetric shallow quantum well. The effective mass renormalized by relativistic corrections and Bychkov–Rashba coefficients are analytically obtained for both cases. It is demonstrated that the general solution transforms to the particular solutions, found previously (Eremko et al., 2015) with the use of spin invariants. The general solution allows to establish conditions at which a specific (accompanied or non-accompanied by Rashba splitting) spin state can be realized. These results can prompt the ways to control the spin degree of freedom via the synthesis of spintronic heterostructures with the required properties.

  2. Hamiltonian structure for two-dimensional extended Green-Naghdi equations

    Matsuno, Yoshimasa


    The two-dimensional Green-Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original Green-Naghdi (GN) model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

  3. Positioning in a flat two-dimensional space-time: the delay master equation

    Coll, Bartolomé; Morales-Lladosa, Juan Antonio


    The basic theory on relativistic positioning systems in a two-dimensional space-time has been presented in two previous papers [Phys. Rev. D {\\bf 73}, 084017 (2006); {\\bf 74}, 104003 (2006)], where the possibility of making relativistic gravimetry with these systems has been analyzed by considering specific examples. Here we study generic relativistic positioning systems in the Minkowski plane. We analyze the information that can be obtained from the data received by a user of the positioning system. We show that the accelerations of the emitters and of the user along their trajectories are determined by the sole knowledge of the emitter positioning data and of the acceleration of only one of the emitters. Moreover, as a consequence of the so called master delay equation, the knowledge of this acceleration is only required during an echo interval, i.e., the interval between the emission time of a signal by an emitter and its reception time after being reflected by the other emitter. We illustrate these result...

  4. One-equation sub-grid scale (SGS) modelling for Euler-Euler large eddy simulation (EELES) of dispersed bubbly flow

    Niceno, B.; Dhotre, M.T.; Deen, N.G.


    In this work, we have presented a one-equation model for sub-grid scale (SGS) kinetic energy and applied it for an Euler-Euler large eddy simulation (EELES) of a bubble column reactor. The one-equation model for SGS kinetic energy shows improved predictions over the state-of-the-art dynamic

  5. A Fibonacci collocation method for solving a class of Fredholm–Volterra integral equations in two-dimensional spaces

    Farshid Mirzaee


    Full Text Available In this paper, we present a numerical method for solving two-dimensional Fredholm–Volterra integral equations (F-VIE. The method reduces the solution of these integral equations to the solution of a linear system of algebraic equations. The existence and uniqueness of the solution and error analysis of proposed method are discussed. The method is computationally very simple and attractive. Finally, numerical examples illustrate the efficiency and accuracy of the method.

  6. Solution and Stability of Euler-Lagrange-Rassias Quartic Functional Equations in Various Quasinormed Spaces

    Heejeong Koh


    Full Text Available We obtain the general solution of Euler-Lagrange-Rassias quartic functional equation of the following . We also prove the Hyers-Ulam-Rassias stability in various quasinormed spaces when .

  7. Accuracy of AFM force distance curves via direct solution of the Euler-Bernoulli equation

    Eppell, Steven J; Liu, Yehe; Zypman, Fredy R


    In an effort to improve the accuracy of force-separation curves obtained from atomic force microscope data, we compare force-separation curves computed using two methods to solve the Euler-Bernoulli equation...

  8. A comparison of SPH schemes for the compressible Euler equations

    Puri, Kunal; Ramachandran, Prabhu


    We review the current state-of-the-art Smoothed Particle Hydrodynamics (SPH) schemes for the compressible Euler equations. We identify three prototypical schemes and apply them to a suite of test problems in one and two dimensions. The schemes are in order, standard SPH with an adaptive density kernel estimation (ADKE) technique introduced Sigalotti et al. (2008) [44], the variational SPH formulation of Price (2012) [33] (referred herein as the MPM scheme) and the Godunov type SPH (GSPH) scheme of Inutsuka (2002) [12]. The tests investigate the accuracy of the inviscid discretizations, shock capturing ability and the particle settling behavior. The schemes are found to produce nearly identical results for the 1D shock tube problems with the MPM and GSPH schemes being the most robust. The ADKE scheme requires parameter values which must be tuned to the problem at hand. We propose an addition of an artificial heating term to the GSPH scheme to eliminate unphysical spikes in the thermal energy at the contact discontinuity. The resulting modification is simple and can be readily incorporated in existing codes. In two dimensions, the differences between the schemes is more evident with the quality of results determined by the particle distribution. In particular, the ADKE scheme shows signs of particle clumping and irregular motion for the 2D strong shock and Sedov point explosion tests. The noise in particle data is linked with the particle distribution which remains regular for the Hamiltonian formulations (MPM and GSPH) and becomes irregular for the ADKE scheme. In the interest of reproducibility, we make available our implementation of the algorithms and test problems discussed in this work.

  9. Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method

    H. S. Shukla


    Full Text Available In this paper, a numerical solution of two dimensional nonlinear coupled viscous Burger equation is discussed with appropriate initial and boundary conditions using the modified cubic B-spline differential quadrature method. In this method, the weighting coefficients are computed using the modified cubic B-spline as a basis function in the differential quadrature method. Thus, the coupled Burger equation is reduced into a system of ordinary differential equations. An optimal five stage and fourth-order strong stability preserving Runge–Kutta scheme is applied for solving the resulting system of ordinary differential equations. The accuracy of the scheme is illustrated by taking two numerical examples. Computed results are compared with the exact solutions and other results available in literature. Obtained numerical result shows that the described method is efficient and reliable scheme for solving two dimensional coupled viscous Burger equation.


    WANG Yue-peng


    The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local (exact) solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.

  11. Global Existence of Smooth Solutions of Compressible Bipolar Euler-Maxwell Equations

    XU Qian-jin; LI Xin; FENG Yue-hong


    The bipolar compressible Euler-Maxwell equations as a fluid dynamic model arising from plasma physics to describe the dynamics of the compressible electrons and ions is investigated.This work is concerned with three-dimensional Euler-Maxwell equations with smooth periodic solutions.With the help of the symmetry operator techniques and energy method,the global smooth solution with small amplitude is constructed around a constant equilibrium solution with asymptotic stability property.

  12. Analytical solution of a multidimensional Langevin equation at high friction limits and probability passing over a two-dimensional saddle

    XING Yong-Zhong


    The analytical solution of a multidimensional Langevin equation at the overdamping limit is obtained and the probability of particles passing over a two-dimensional saddle point is discussed. These results may break a path for studying further the fusion in superheavy elements synthesis.

  13. Improvement of Convergence to Steady State Solutions of Euler Equations with Weighted Compact Nonlinear Schemes

    Shu-hai ZHANG; Xiao-gang DENG; Mei-liang MAO; Chi-Wang SHU


    The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) [Deng X.and Zhang H.(2000),J.Comput.Phys.165,22-44 and Zhang S.,Jiang S.and Shu C.-W.(2008),J.Comput.Phys.227,7294-7321] is studied through numerical tests.Like most other shock capturing schemes,WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level.In this paper,the techniques studied in [Zhang S.and Shu.C.-W.(2007),J.Sci.Comput.31,273-305 and Zhang S.,Jiang S and Shu.C.-W.(2011),J.Sci.Comput.47,216-238],to improve the convergence to steady state solutions for WENO schemes,are generalized to the WCNS.Detailed numerical studies in one and two dimensional cases are performed.Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS.The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.

  14. Evaluating a linearized Euler equations model for strong turbulence effects on sound propagation.

    Ehrhardt, Loïc; Cheinet, Sylvain; Juvé, Daniel; Blanc-Benon, Philippe


    Sound propagation outdoors is strongly affected by atmospheric turbulence. Under strongly perturbed conditions or long propagation paths, the sound fluctuations reach their asymptotic behavior, e.g., the intensity variance progressively saturates. The present study evaluates the ability of a numerical propagation model based on the finite-difference time-domain solving of the linearized Euler equations in quantitatively reproducing the wave statistics under strong and saturated intensity fluctuations. It is the continuation of a previous study where weak intensity fluctuations were considered. The numerical propagation model is presented and tested with two-dimensional harmonic sound propagation over long paths and strong atmospheric perturbations. The results are compared to quantitative theoretical or numerical predictions available on the wave statistics, including the log-amplitude variance and the probability density functions of the complex acoustic pressure. The match is excellent for the evaluated source frequencies and all sound fluctuations strengths. Hence, this model captures these many aspects of strong atmospheric turbulence effects on sound propagation. Finally, the model results for the intensity probability density function are compared with a standard fit by a generalized gamma function.

  15. Coupling Navier-stokes and Cahn-hilliard Equations in a Two-dimensional Annular flow Configuration

    Vignal, Philippe


    In this work, we present a novel isogeometric analysis discretization for the Navier-Stokes- Cahn-Hilliard equation, which uses divergence-conforming spaces. Basis functions generated with this method can have higher-order continuity, and allow to directly discretize the higher- order operators present in the equation. The discretization is implemented in PetIGA-MF, a high-performance framework for discrete differential forms. We present solutions in a two- dimensional annulus, and model spinodal decomposition under shear flow.

  16. Correction to Euler's equations and elimination of the closure problem in turbulence

    Zak, Michail


    It has been demonstrated that the Euler equations of inviscid fluid are incomplete: according to the principle of release of constraints, absence of shear stresses must be compensated by additional degrees of freedom, and leads to Reynolds-type multivalued velocity field. however unlike the Reynolds equations, the enlarged Euler's (EE) model provides additional equations for fluctuations, and that eliminates the closure problem. Therefore the (EE) equations are applicable to fully developed turbulent motions where the physical viscosity is vanishingly small compare to the turbulent viscosity, as well as to superfluids and atomized fluids.Analysis of coupled mean/fluctuation EE equations shows that fluctuations stabilize the whole system generating elastic shear waves and increasing speed of sound. Those turbulent motions that originated from instability of underlying laminar motions can be described by the modified Euler's equation with the closure provided by the stabilization principle: driven by instabilit...

  17. Analysis of Lagrange's original derivation of the Euler-Lagrange Differential Equation

    Laughlin, Ryan; Close, Hunter


    The Euler-Lagrange differential equation provides the Lagrangian equations of motion, and thus allows the exact trajectory of an object in a potential to be found. We analyze the original derivation of the Euler-Lagrange differential equation via a translation of the third edition of Lagrange's Mecanique Analytique (1811). We compare and contrast this derivation with the derivation commonly done in a junior-level classical mechanics course. Lagrange uses several founding concepts to produce a generalized equation of motion for all dynamics. These concepts are, in the order addressed by Lagrange, the Principle of Virtual Velocities, the Conservation des Forces Vives, and the Principle of Least Action. Lagrange then employs what he calls the Method of Variations to the general equation of motion for dynamics to ultimately resolve something similar to the Euler-Lagrange Differential equation we use today. We also compare modern notation with Lagrange's notation.

  18. Exact solutions of the two-dimensional discrete nonlinear Schrodinger equation with saturable nonlinearity

    Khare, A.; Rasmussen, K. O.; Samuelsen, Mogens Rugholm


    We show that the two-dimensional, nonlinear Schrodinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the e......We show that the two-dimensional, nonlinear Schrodinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show...

  19. An improved front tracking method for the Euler equations

    J.A.S. Witteveen (Jeroen); B. Koren (Barry); P.G. Bakker


    textabstractAn improved front tracking method for hyperbolic conservation laws is presented. The improved method accurately resolves discontinuities as well as continuous phenomena. The method is based on an improved front interaction model for a physically more accurate modeling of the Euler

  20. Orthonormal quaternion frames, Lagrangian evolution equations, and the three-dimensional Euler equations

    Gibbon, John


    More than 160 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the orientation and paths of moving objects undergoing three-axis rotations. Here it is shown that they provide a natural way of selecting an appropriate orthonormal frame—designated the quaternion-frame—for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid equations is then considered. This work has some bearing on the issue of whether the Euler equations develop a singularity in a finite time. Some of the literature on this topic is reviewed, which includes both the Beale-Kato-Majda theorem and associated work on the direction of vorticity by Constantin, Fefferman, and Majda and by Deng, Hou, and Yu. It is then shown how the quaternion formalism provides an alternative formulation in terms of the Hessian of the pressure.

  1. Exact Solutions of the Two-Dimensional Discrete Nonlinear Schr\\"odinger Equation with Saturable Nonlinearity

    Khare, Avinash; Samuelsen, Mogens R; Saxena, Avadh; 10.1088/1751-8113/43/37/375209


    We show that the two-dimensional, nonlinear Schr\\"odinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero.

  2. An instability mechanism of pulsatile flow along particle trajectories for the axisymmetric Euler equations

    Yoneda, Tsuyoshi


    The dynamics along the particle trajectories for the 3D axisymmetric Euler equations in an infinite cylinder are considered. It is shown that if the inflow-outflow is rapidly increasing in time, the corresponding laminar profile of the Euler flow is not (in some sense) stable provided that the swirling component is not small. This exhibits an instability mechanism of pulsatile flow. In the proof, Frenet-Serret formulas and orthonormal moving frame are essentially used.

  3. Numerical method of the Riemann problem for two-dimensional multi-fluid flows with general equation of state

    Bai Jing-Song; Zhang Zhan-Ji; Li Ping; Zhong Min


    Based on the classical Roe method, we develop an interface capture method according to the general equation of state, and extend the single-fluid Roe method to the two-dimensional (2D) multi-fluid flows, as well as construct the continuous Roe matrix for the whole flow field. The interface capture equations and fluid dynamic conservative equations are coupled together and solved by using any high-resolution schemes that usually suit for the single-fluid flows. Some numerical examples are given to illustrate the solution of 1D and 2D multi-fluid Riemann problems.

  4. Global well-posedness of strong solutions to the two-dimensional barotropic compressible Navier-Stokes equations with vacuum

    Fang, Li; Guo, Zhenhua


    The aim of this paper is to establish the global well-posedness and large-time asymptotic behavior of the strong solution to the Cauchy problem of the two-dimensional compressible Navier-Stokes equations with vacuum. It is proved that if the shear viscosity {μ} is a positive constant and the bulk viscosity {λ} is the power function of the density, that is, {λ=ρ^{β}} with {β in [0,1],} then the Cauchy problem of the two-dimensional compressible Navier-Stokes equations admits a unique global strong solution provided that the initial data are of small total energy. This result can be regarded as the extension of the well-posedness theory of classical compressible Navier-Stokes equations [such as Huang et al. (Commun Pure Appl Math 65:549-585, 2012) and Li and Xin ( respectively]. Furthermore, the large-time behavior of the strong solution to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations had been also obtained.

  5. Error transport equation boundary conditions for the Euler and Navier-Stokes equations

    Phillips, Tyrone S.; Derlaga, Joseph M.; Roy, Christopher J.; Borggaard, Jeff


    Discretization error is usually the largest and most difficult numerical error source to estimate for computational fluid dynamics, and boundary conditions often contribute a significant source of error. Boundary conditions are described with a governing equation to prescribe particular behavior at the boundary of a computational domain. Boundary condition implementations are considered sufficient when discretized with the same order of accuracy as the primary governing equations; however, careless implementations of boundary conditions can result in significantly larger numerical error. Investigations into different numerical implementations of Dirichlet and Neumann boundary conditions for Burgers' equation show a significant impact on the accuracy of Richardson extrapolation and error transport equation discretization error estimates. The development of boundary conditions for Burgers' equation shows significant improvements in discretization error estimates in general and a significant improvement in truncation error estimation. The latter of which is key to accurate residual-based discretization error estimation. This research investigates scheme consistent and scheme inconsistent implementations of inflow and outflow boundary conditions up to fourth order accurate and a formulation for a slip wall boundary condition for truncation error estimation are developed for the Navier-Stokes and Euler equations. The scheme consistent implementation resulted in much smoother truncation error near the boundaries and more accurate discretization error estimates.

  6. Physics-based Stabilization of Spectral Elements for the 3D Euler Equations of Moist Atmospheric Convection


    3D Euler Equations of Moist Atmospheric Convection 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER...STABILIZATION OF SPECTRAL ELEMENTS FOR THE 3D EULER EQUATIONS OF MOIST ATMOSPHERIC CONVECTION SIMONE MARRAS, ANDREAS MÜLLER, FRANCIS X. GIRALDO Dept. Appl...spectral elements, we introduce a dissipative scheme based on the solution of the compressible Euler equations that are regularized through the addi

  7. First Characterization of a New Method for Numerically Solving the Dirichlet Problem of the Two-Dimensional Electrical Impedance Equation

    Marco Pedro Ramirez-Tachiquin; Cesar Marco Antonio Robles Gonzalez; Rogelio Adrian Hernandez-Becerril; Ariana Guadalupe Bucio Ramirez


    Based upon the elements of the modern pseudoanalytic function theory, we analyze a new method for numerically solving the forward Dirichlet boundary value problem corresponding to the two-dimensional electrical impedance equation. The analysis is performed by introducing interpolating piecewise separable-variables conductivity functions in the unit circle. To warrant the effectiveness of the posed method, we consider several examples of conductivity functions, whose boundary condi...

  8. The Use of Iterative Methods to Solve Two-Dimensional Nonlinear Volterra-Fredholm Integro-Differential Equations

    shadan sadigh behzadi


    Full Text Available In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integro-differential equation by using the following powerful, efficient but simple methods: (i Modified Adomian decomposition method (MADM, (ii Variational iteration method (VIM, (iii Homotopy analysis method (HAM and (iv Modified homotopy perturbation method (MHPM. The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate the accuracy of the presented methods.

  9. A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation

    Sun, HongGuang; Liu, Xiaoting; Zhang, Yong; Pang, Guofei; Garrard, Rhiannon


    Fractional-order diffusion equations (FDEs) extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media. Real-world diffusion can be multi-dimensional, requiring efficient numerical solvers that can handle long-term memory embedded in mass transport. To address this challenge, a semi-discrete Kansa method is developed to approximate the two-dimensional spatiotemporal FDE, where the Kansa approach first discretizes the FDE, then the Gauss-Jacobi quadrature rule solves the corresponding matrix, and finally the Mittag-Leffler function provides an analytical solution for the resultant time-fractional ordinary differential equation. Numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes. Applications further show that the numerical method can efficiently solve two-dimensional spatiotemporal FDE models with either a continuous or discrete mixing measure. Hence this study provides an efficient and fast computational method for modeling super-diffusive, sub-diffusive, and mixed diffusive processes in large, two-dimensional domains with irregular shapes.

  10. Large Scale Simulations of the Euler Equations on GPU Clusters

    Liebmann, Manfred


    The paper investigates the scalability of a parallel Euler solver, using the Vijayasundaram method, on a GPU cluster with 32 Nvidia Geforce GTX 295 boards. The aim of this research is to enable large scale fluid dynamics simulations with up to one billion elements. We investigate communication protocols for the GPU cluster to compensate for the slow Gigabit Ethernet network between the GPU compute nodes and to maintain overall efficiency. A diesel engine intake-port and a nozzle, meshed in different resolutions, give good real world examples for the scalability tests on the GPU cluster. © 2010 IEEE.

  11. Solving Two -Dimensional Diffusion Equations with Nonlocal Boundary Conditions by a Special Class of Padé Approximants

    Mohammad Siddique


    Full Text Available Parabolic partial differential equations with nonlocal boundary conditions arise in modeling of a wide range of important application areas such as chemical diffusion, thermoelasticity, heat conduction process, control theory and medicine science. In this paper, we present the implementation of positivity- preserving Padé numerical schemes to the two-dimensional diffusion equation with nonlocal time dependent boundary condition. We successfully implemented these numerical schemes for both Homogeneous and Inhomogeneous cases. The numerical results show that these Padé approximation based numerical schemes are quite accurate and easily implemented.

  12. Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

    Fukang Yin


    Full Text Available A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs. The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs. The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

  13. A Finite-Element Solution of the Navier-Stokes Equations for Two-Dimensional and Axis-Symmetric Flow

    Sven Ø. Wille


    Full Text Available The finite element formulation of the Navier-Stokes equations is derived for two-dimensional and axis-symmetric flow. The simple triangular, T6, isoparametric element is used. The velocities are interpolated by quadratic polynomials and the pressure is interpolated by linear polynomials. The non-linear simultaneous equations are solved iteratively by the Newton-Raphson method and the element matrix is given in the Newton-Raphson form. The finite element domain is organized in substructures and an equation solver which works on each substructure is specially designed. This equation solver needs less storage in the computer and is faster than the traditional banded equation solver. To reduce the amount of input data an automatic mesh generator is designed. The input consists of the coordinates of eight points defining each substructure with the corresponding boundary conditions. In order to interpret the results they are plotted on a calcomp plotter. Examples of plots of the velocities, the streamlines and the pressure inside a two-dimensional flow divider and an axis-symmetric expansion of a tube are shown for various Reynolds numbers.

  14. Asymptotic solution for high vorticity regions in incompressible 3D Euler equations

    Agafontsev, D S; Mailybaev, A A


    Incompressible 3D Euler equations develop high vorticity in very thin pancake-like regions from generic large-scale initial conditions. In this work we propose an exact solution of the Euler equations for the asymptotic pancake evolution. This solution combines a shear flow aligned with an asymmetric straining flow, and is characterized by a single asymmetry parameter and an arbitrary transversal vorticity profile. The analysis is based on detailed comparison with numerical simulations performed using a pseudo-spectral method in anisotropic grids of up to 972 x 2048 x 4096.

  15. Admissibility of weak solutions for the compressible Euler equations, n ≥ 2.

    Slemrod, Marshall


    This paper compares three popular notions of admissibility for weak solutions of the compressible isentropic Euler equations of gas dynamics: (i) the viscosity criterion, (ii) the entropy inequality (the thermodynamically admissible isentropic solutions), and (iii) the viscosity-capillarity criterion. An exact summation of the Chapman-Enskog expansion for Grad's moment system suggests that it is the third criterion that is representing the kinetic theory of gases. This, in turn, may shed some light on the ability to recover weak solutions of the Euler equations via a hydrodynamic limit.

  16. Pseudo-time method for optimal shape design using the Euler equations

    Iollo, Angelo; Kuruvila, Geojoe; Ta'asan, Shlomo


    We exploit a novel idea for the optimization of flows governed by the Euler equations. The algorithm consists of marching on the design hypersurface while improving the distance to the state and costate hypersurfaces. We consider the problem of matching the pressure distribution to a desired one, subject to the euler equations, both for subsonic and supersonic flows. The rate of convergence to the minimum for the cases considered is 3 to 4 times slower than that of the analysis problem. Results are given for Ringleb flow and a shockless recompression case.

  17. Eigenmode Analysis of Boundary Conditions for One-Dimensional Preconditioned Euler Equations

    Darmofal, David L.


    An analysis of the effect of local preconditioning on boundary conditions for the subsonic, one-dimensional Euler equations is presented. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective with preconditioning, and, at low Mach numbers, disturbances do not decay. Other boundary conditions are investigated which are non-reflective with preconditioning and numerical results are presented confirming the analysis.


    Hua-zhong Tang


    This paper attempts to develop kinetic flux vector splitting (KFVS) for the Euler equations with general pressure laws. It is well known that the gas distribution function for the local equilibrium state plays an important role in the construction of the gas-kinetic schemes. To recover the Euler equations with a general equation of state (EOS), a new local equilibrium distribution is introduced with two parameters of temperature approximation decided uniquely by macroscopic variables. Utilizing the well-known connection that the Euler equations of motion are the moments of the Boltzmann equation whenever the velocity distribution function is a local equilibrium state, a class of high resolution MUSCL-type KFVS schemes are presented to approximate the Euler equations of gas dynamics with a general EOS. The schemes are finally applied to several test problems for a general EOS. In comparison with the exact solutions, our schemes give correct location and more accurate resolution of discontinuities. The extension of our idea to multidimensional case is natural.

  19. A correction on two dimensional KdV equation with topography

    XU Zhaoting; Efim PELINOVSKY; SHEN Guojin; Tapiana TALIPOVA


    The correction on the 2D KdV equation derived by Djordjevic and Redekopp is presented. A lapsus calami in the 2D KdV equation is removed by means of the conservation principle of the energy flux in a wave ray tube. The results show that the coefficient of the third term in the inhomogeneous term of 2D KdV equation in the paper of Djordjevic and Redekopp is 2, instead of 3.

  20. On the rotational equations of motion in rigid body dynamics when using Euler parameters.

    Sherif, Karim; Nachbagauer, Karin; Steiner, Wolfgang

    Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and it will be shown that they can be transformed into each other. Special attention is hereby given to the value of the Lagrange multiplier and the complexity of terms representing the inertia forces. Particular attention is also paid to the rotational generalized external force vector, which is not unique when using Euler parameters as rotational coordinates.

  1. Initial and Boundary Value Problems for Two-Dimensional Non-hydrostatic Boussinesq Equations

    沈春; 孙梅娜


    Based on the theory of stratification, the weU-posedness of the initial and boundary value problems for the system of twodimensional non-hydrostatic Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for some representative initial and boundary value problems. Several special cases were discussed.

  2. Construction of Green's Functions for the Two-Dimensional Static Klein-Gordon Equation



    In contrast to the cognate Laplace equation, for which a vast number of Green's functions is available, the field is not that developed for the static Klein-Gordon equation. The latter represents, nonetheless, a natural area for application of some of the methods that are proven productive for the Laplace equation. The perspective looks especially attractive for the methods of images and eigenfunction expansion.This study is based on our experience recently gained on the construction of Green's functions for elliptic partial differential equations. An extensive list of boundary-value problems formulated for the static Klein-Gordon equation is considered. Computerfriendly representations of their Green's functions are obtained, most of which have never been published before.

  3. Gravity and Zonal Flows of Giant Planets: From the Euler Equation to the Thermal Wind Equation

    Cao, Hao


    Any non-spherical distribution of density inside planets and stars gives rise to a non-spherical external gravity and change of shape. If part or all of the observed zonal flows at the cloud deck of giant planets represent deep interior dynamics, then the density perturbations associated with the deep zonal flows could generate gravitational signals detectable by the planned Juno mission and the Cassini Proximal Orbits. It is currently debated whether the thermal wind equation (TWE) can be used to calculate the gravity field associated with deep zonal flows. Here we present a critical comparison between the Euler equation and the thermal wind equation. Our analysis shows that the applicability of the TWE in calculating the gravity moments depends crucially on retaining the non-sphericity of the background density and gravity. Only when the background non-sphericity of the planet is taken into account, the TWE makes accurate enough prediction (with a few tens of percent errors) for the high-degree gravity mome...


    Benyu Guo; Songnian He; Heping Ma


    A mixed Chebyshev spectral-finite element method is proposed for solving two-dimensionalunsteady Navier-Stokes equation. The generalized stability and convergence are proved.The numerical results show the advantages of this method.

  5. Two-Dimensional Saddle Point Equation of Ginzburg-Landau Hamiltonian for the Diluted Ising Model

    WU Xin-Tian


    @@ The saddle point equation of Ginzburg-Landau Hamiltonian for the diluted Ising model is developed. The ground state is solved numerically in two dimensions. The result is partly explained by the coarse-grained approximation.

  6. From Discreteness to Continuity: Dislocation Equation for Two-Dimensional Triangular Lattice

    WANG Shao-Feng


    @@ A systematic method from the discreteness to the continuity is presented for the dislocation equation of the triangular lattice. A modification of the Peierls equation has been derived strictly. The modified equation includes the higher order corrections of the discrete effect which are important for the core structure of dislocation. It is observed that the modified equation possesses a universal form which is model-independent except the factors.The factors, which depend on the detail of the model, are related to the derivatives of the kernel at its zero point in the wave-vector space. The results open a way to deal with the complicated models because what one needs to do is to investigate the behaviour near the zero point of the kernel in the wave-vector space instead of calculating the kernel completely.

  7. Two Dimensional Cahn-Hilliard Equation with Concentration Dependent Mobility and Gradient Dependent Potential



    In this paper we consider the initial boundary value problem of CahnHilliard equation with concentration dependent mobility and gradient dependent potential. By the Lp type estimates and the theory of Morrey spaces, we prove the H(o)der continuity of the solutions. Then we obtain the existence of global classical solutions. The present work can be viewed as an extension to the previous work on the Cahn-Hilliard equation with concentration dependent mobility and potential.

  8. An equation for pressure of a two-dimensional Yukawa liquid

    Feng, Yan; Li, Wei; Wang, Qiaoling; Lin, Wei; Goree, John; Liu, Bin


    Thermodynamic behavior of two-dimensional (2D) dusty plasmas has been studied experimentally and theoretically recently. As a crucial parameter in thermodynamics, the pressure of dusty plasmas arises from frequent collisions of individual dust particles. Here, equilibrium molecular dynamical simulations were performed to study the pressure of 2D Yukawa liquids. A simple analytical expression for the pressure of a 2D Yukawa liquid is found by fitting the obtained pressure data over a wide range of temperatures, from the coldest close to the melting point, to the hottest about 70 times higher than the melting points. The obtained expression verifies that the pressure can be written as the sum of a potential term which is a simple multiple of the Coulomb potential energy at a distance of Wigner-Seitz radius, and a kinetic term which is a multiple of the one for an ideal gas. Dimensionless coefficients for each of these terms are found empirically, by fitting. The resulting analytical expression, with its empirically determined coefficients, is plotted as isochors, or curves of constant area. These results should be applicable to 2D dusty plasmas. Work in China supported by by the National Natural Science Foundation of China under Grant No. 11505124, the 1000 Youth Talents Plan, and startup funds from Soochow University. Work in the US supported by DOE & NSF.

  9. Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations: Two-dimensional case.

    Zhang, Jiwei; Xu, Zhenli; Wu, Xiaonan


    This paper aims to design local absorbing boundary conditions (LABCs) for the two-dimensional nonlinear Schrödinger equations on a rectangle by extending the unified approach. Based on the time-splitting idea, the main process of the unified approach is to approximate the kinetic energy part by a one-way equation, unite it with the potential energy equation, and then obtain the well-posed and accurate LABCs on the artificial boundaries. In the corners, we use the (1,1)-Padé approximation to the kinetic term and also unite it with the nonlinear term to give some local corner boundary conditions. Numerical tests are given to verify the stable and tractable advantages of the method.

  10. Entropy Analysis of Kinetic Flux Vector Splitting Schemes for the Compressible Euler Equations

    Shiuhong, Lui; Xu, Jun


    Flux Vector Splitting (FVS) scheme is one group of approximate Riemann solvers for the compressible Euler equations. In this paper, the discretized entropy condition of the Kinetic Flux Vector Splitting (KFVS) scheme based on the gas-kinetic theory is proved. The proof of the entropy condition involves the entropy definition difference between the distinguishable and indistinguishable particles.

  11. Risk of Rare Disasters, Euler Equation Errors and the Performance of the C-CAPM

    Posch, Olaf; Schrimpf, Andreas

    This paper shows that the consumption-based asset pricing model (C-CAPM) with low-probability disaster risk rationalizes large pricing errors, i.e. Euler equation errors. This result is remarkable, since Lettau and Ludvigson (2009) show that leading asset pricing models cannot explain sizeable...

  12. Euler-Lagrange Equations for the Gribov Reggeon Calculus in QCD and in Gravity

    Lipatov, L. N.

    The theory of the high energy scattering in QCD and gravity is based on the reggeization of gluons and gravitons, respectively. We discuss the corresponding effective actions for reggeized particle interactions. The Euler-Lagrange equations in these theories are constructed with a variational approach for the effective actions and by using their invariance under the gauge and general coordinate transformations.

  13. Asymptotic Behavior of Global Solution for Nonlinear Generalized Euler-Possion-Darboux Equation

    LIANGBao-song; CHENZhen


    J. L Lions and W. A. Stranss [1] have proved the existence of a global solution of the initial boundary value problem for nonlinear generalized Euler-Possion-Darboux equation. In this paper we are going to investigate the asymptotic behavior of the global solution by a difference inequality.

  14. A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids

    Pesch, L.; van der Vegt, Jacobus J.W.


    Using the generalized variable formulation of the Euler equations of fluid dynamics, we develop a numerical method that is capable of simulating the flow of fluids with widely differing thermodynamic behavior: ideal and real gases can be treated with the same method as an incompressible fluid. The

  15. A dynamic multiblock approach to solving the unsteady Euler equations about complex configurations

    Arabshahi, Abdollah

    The objective is the development of a numerical method which can accurately and economically solve the unsteady Euler equations for three-dimensional flow fields around complex configurations, particularly a generic aircraft with a store in the captive and vertical launch position. A cell centered finite volume spatial discretization is applied to the three-dimensional, time-dependent, Euler equations written in general time-dependent curvilinear coordinates. Two algorithms are presented for solving the system of Euler equations. The first algorithm is based on flux-vector splitting while the second algorithm is based on flux-difference splitting using Roe averaged variables. For both algorithms, an implicit upwind biased approach is employed to integrate the spatially discretized equations in time. The multiblock technique utilizes the concept of decomposing the flow field between the surfaces of the configuration and some outer far field boundary into a set of blocks. Calculated results compared with experimental data indicate that the present Euler solver can calculate transonic flow fields efficiently and accurately over complex geometries. Furthermore, the results demonstrate how computational fluid dynamics (CFD) can be used to accurately simulate steady and, for the first time, unsteady fluid flow over a complete wing-pylon-store configuration with the store in the captive and vertical launch positions.

  16. A meshless front tracking method for the Euler equations of fluid dynamics

    Witteveen, J.A.S.


    A second order front tracking method is developed for solving the Euler equations of inviscid fluid dynamics numerically. Front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here the second order convergence



    Using the integral average technique and a new function,some new oscillation criteria related to Euler integral are obtained for second order nonlinear differential equations with damping and forcing. Our results are of a higher degree of generality than some previous results. Information about the distribution of the zeros of solutions to the system is also obtained.

  18. A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids

    Pesch, L.; Vegt, van der J.J.W.


    Using the generalized variable formulation of the Euler equations of fluid dynamics, we develop a numerical method that is capable of simulating the flow of fluids with widely differing thermodynamic behavior: ideal and real gases can be treated with the same method as an incompressible fluid. The w

  19. Note on a differentiation formula, with application to the two-dimensional Schrödinger equation


    A method for obtaining discretization formulas for the derivatives of a function is presented, which relies on a generalization of divided differences. These modified divided differences essentially correspond to a change of the dependent variable. This method is applied to the numerical solution of the eigenvalue problem for the two-dimensional Schrödinger equation, where standard methods converge very slowly while the approach proposed here gives accurate results. The presented approach has the merit of being conceptually simple and might prove useful in other instances. PMID:28178300

  20. A meshless method using radial basis functions for numerical solution of the two-dimensional KdV-Burgers equation

    Zabihi, F.; Saffarian, M.


    The aim of this article is to obtain the numerical solution of the two-dimensional KdV-Burgers equation. We construct the solution by using a different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. We use a predictor-corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

  1. Vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow

    Bardos, Claude; Wiedemann, Emil


    We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray-Hopf weak solutions for the Navier-Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier-Stokes equations could serve as a uniqueness criterion for weak solutions of the Euler equations.

  2. Euler-Chebyshev methods for integro-differential equations

    Houwen, P.J. van der; Sommeijer, B.P.


    We construct and analyse explicit methods for solving initial value problems for systems of differential equations with expensive righthand side functions whose Jacobian has its stiff eigenvalues along the negative axis. Such equations arise after spatial discretization of parabolic integro-differen

  3. The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations

    Choi, Young-Pil; Kwon, Bongsuk


    We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate L2 Sobolev space. We also establish the a priori large-time behavior estimate by constructing a Lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities, and using this together with the bootstrapping argument, we obtain the global classical solution. The large-time behavior estimate asserts that the velocity functions of the pressureless Euler and the compressible Navier-Stokes equations are aligned exponentially fast as time tends to infinity.

  4. On the Choquet-Bruhat-York-Friedrich formulation of the Einstein-Euler equations

    Disconzi, Marcelo M


    Short-time existence for the Einstein-Euler and the vacuum Einstein equations is proven using a Friedrich inspired formulation due to Choquet-Bruhat and York, where the system is cast into a symmetric hyperbolic form and the Riemann tensor is treated as one of the fundamental unknowns of the problem. The reduced system of Choquet-Bruhat and York, along with the preservation of the gauge, is shown to imply the full Einstein equations.



    In this paper,the Riemann problem of the 1-D reduced model for the 2-D Euler equations is considered and the Riemann solutions are obtained.It is proved that,as the pressure vanishes,they converge to two kinds of Riemann solutions to the 1D reduced model for the 2-D transport equations:one contains δ-shocks,the other contains vacuum.

  6. Field theory and weak Euler-Lagrange equation for classical particle-field systems.

    Qin, Hong; Burby, Joshua W; Davidson, Ronald C


    It is commonly believed as a fundamental principle that energy-momentum conservation of a physical system is the result of space-time symmetry. However, for classical particle-field systems, e.g., charged particles interacting through self-consistent electromagnetic or electrostatic fields, such a connection has only been cautiously suggested. It has not been formally established. The difficulty is due to the fact that the dynamics of particles and the electromagnetic fields reside on different manifolds. We show how to overcome this difficulty and establish the connection by generalizing the Euler-Lagrange equation, the central component of a field theory, to a so-called weak form. The weak Euler-Lagrange equation induces a new type of flux, called the weak Euler-Lagrange current, which enters conservation laws. Using field theory together with the weak Euler-Lagrange equation developed here, energy-momentum conservation laws that are difficult to find otherwise can be systematically derived from the underlying space-time symmetry.

  7. Spin eigen-states of Dirac equation for quasi-two-dimensional electrons

    Eremko, Alexander, E-mail: [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); Brizhik, Larissa, E-mail: [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); Loktev, Vadim, E-mail: [Bogolyubov Institute for Theoretical Physics, Metrologichna Sttr., 14-b, Kyiv, 03680 (Ukraine); National Technical University of Ukraine “KPI”, Peremohy av., 37, Kyiv, 03056 (Ukraine)


    Dirac equation for electrons in a potential created by quantum well is solved and the three sets of the eigen-functions are obtained. In each set the wavefunction is at the same time the eigen-function of one of the three spin operators, which do not commute with each other, but do commute with the Dirac Hamiltonian. This means that the eigen-functions of Dirac equation describe three independent spin eigen-states. The energy spectrum of electrons confined by the rectangular quantum well is calculated for each of these spin states at the values of energies relevant for solid state physics. It is shown that the standard Rashba spin splitting takes place in one of such states only. In another one, 2D electron subbands remain spin degenerate, and for the third one the spin splitting is anisotropic for different directions of 2D wave vector.

  8. Numerical computation of the critical energy constant for two-dimensional Boussinesq equations

    Kolkovska, N.; Angelow, K.


    The critical energy constant is of significant interest for the theoretical and numerical analysis of Boussinesq type equations. In the one-dimensional case this constant is evaluated exactly. In this paper we propose a method for numerical evaluation of this constant in the multi-dimensional cases by computing the ground state. Aspects of the numerical implementation are discussed and many numerical results are demonstrated.

  9. The Difference Format of Landau-Lifshitz Equation in Two-dimensional Case

    Zhong Taiyong


    Full Text Available In this paper, the author considers a difference scheme of Laudau-Lifshitz equation (LL for short and modulus of unj which are constantly remaining equal to 1. Using this iteration format error which is ordered to t/2h2 , the author comes to a conclusion based on several initial simulations. According to some conditions, the author gives the numerical solution, the examples of exact solution and the error comparisons of the solutions.

  10. MARG2D code. 1. Eigenvalue problem for two dimensional Newcomb equation

    Tokuda, Shinji [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Watanabe, Tomoko


    A new method and a code MARG2D have been developed to solve the 2-dimensional Newcomb equation which plays an important role in the magnetohydrodynamic (MHD) stability analysis in an axisymmetric toroidal plasma such as a tokamak. In the present formulation, an eigenvalue problem is posed for the 2-D Newcomb equation, where the weight function (the kinetic energy integral) and the boundary conditions at rational surfaces are chosen so that an eigenfunction correctly behaves as the linear combination of the small solution and the analytical solutions around each of the rational surfaces. Thus, the difficulty on solving the 2-D Newcomb equation has been resolved. By using the MARG2D code, the ideal MHD marginally stable state can be identified for a 2-D toroidal plasma. The code is indispensable on computing the outer-region matching data necessary for the resistive MHD stability analysis. Benchmark with ERATOJ, an ideal MHD stability code, has been carried out and the MARG2D code demonstrates that it indeed identifies both stable and marginally stable states against ideal MHD motion. (author)

  11. Correction to Euler's equations and elimination of the closure problem in turbulence

    Michail Zak


    Full Text Available It has been demonstrated that the Euler equations of inviscid fluid are incomplete: according to the principle of release of constraints, absence of shear stresses must be compensated by additional degrees of freedom, and that leads to a Reynolds-type multivalued velocity field. However, unlike the Reynolds equations, the enlarged Euler's (EE model provides additional equations for fluctuations, and that eliminates the closure problem. Therefore the EE equations are applicable to fully developed turbulent motions where the physical viscosity is vanishingly small compare to the turbulent viscosity, as well as to superfluids and atomized fluids. Analysis of coupled mean/fluctuation EE equations shows that fluctuations stabilize the whole system generating elastic shear waves and increasing speed of sound. Those turbulent motions that originated from instability of underlying laminar motions can be described by the modified Euler's equation with the closure provided by the stabilization principle: driven by instability of laminar motion, fluctuations grow until the new state attains a neutral stability in the enlarged (multivalued class of functions, and these fluctuations can be taken as boundary conditions for the EE model. The approach is illustrated by an example.

  12. An adaptive, high-order phase-space remapping for the two-dimensional Vlasov-Poisson equations

    Wang, Bei; Colella, Phil


    The numerical solution of high dimensional Vlasov equation is usually performed by particle-in-cell (PIC) methods. However, due to the well-known numerical noise, it is challenging to use PIC methods to get a precise description of the distribution function in phase space. To control the numerical error, we introduce an adaptive phase-space remapping which regularizes the particle distribution by periodically reconstructing the distribution function on a hierarchy of phase-space grids with high-order interpolations. The positivity of the distribution function can be preserved by using a local redistribution technique. The method has been successfully applied to a set of classical plasma problems in one dimension. In this paper, we present the algorithm for the two dimensional Vlasov-Poisson equations. An efficient Poisson solver with infinite domain boundary conditions is used. The parallel scalability of the algorithm on massively parallel computers will be discussed.

  13. General method and exact solutions to a generalized variable-coefficient two-dimensional KdV equation

    Chen, Yong [Ningbo Univ., Ningbo (China). Department of Mathematics; Shanghai Jiao-Tong Univ., Shangai (China). Department of Physics; Chinese Academy of sciences, Beijing (China). Key Laboratory of Mathematics Mechanization


    A general method to uniformly construct exact solutions in terms of special function of nonlinear partial differential equations is presented by means of a more general ansatz and symbolic computation. Making use of the general method, we can successfully obtain the solutions found by the method proposed by Fan (J. Phys. A., 36 (2003) 7009) and find other new and more general solutions, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solution, soliton solutions, soliton-like solutions and Jacobi, Weierstrass doubly periodic wave solutions. A general variable-coefficient two-dimensional KdV equation is chosen to illustrate the method. As a result, some new exact soliton-like solutions are obtained. planets. The numerical results are given in tables. The results are discussed in the conclusion.

  14. First characterization of a new method for numerically solving the Dirichlet problem of the two-dimensional Electrical Impedance Equation

    T., M P Ramirez; Hernandez-Becerril, R A


    Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation. The analysis is performed by interpolating piecewise separable-variables conductivity functions, that are eventually used in the numerical calculations in order to obtain finite sets of orthonormal functions, whose linear combinations succeed to approach the imposed boundary conditions. To warrant the effectiveness of the numerical method, we study six different examples of conductivity. The boundary condition for every case is selected considering one exact solution of the Electrical Impedance Equation. The work intends to discuss the contributions of these results into the field of the Electrical Impedance Tomography.

  15. Two Hybrid Methods for Solving Two-Dimensional Linear Time-Fractional Partial Differential Equations

    B. A. Jacobs


    Full Text Available A computationally efficient hybridization of the Laplace transform with two spatial discretization techniques is investigated for numerical solutions of time-fractional linear partial differential equations in two space variables. The Chebyshev collocation method is compared with the standard finite difference spatial discretization and the absolute error is obtained for several test problems. Accurate numerical solutions are achieved in the Chebyshev collocation method subject to both Dirichlet and Neumann boundary conditions. The solution obtained by these hybrid methods allows for the evaluation at any point in time without the need for time-marching to a particular point in time.

  16. Two-dimensional ultrasound measurement of thyroid gland volume: a new equation with higher correlation with 3-D ultrasound measurement.

    Ying, Michael; Yung, Dennis M C; Ho, Karen K L


    This study aimed to develop a new two-dimensional (2-D) ultrasound thyroid volume estimation equation using three-dimensional (3-D) ultrasound as the standard of reference, and to compare the thyroid volume estimation accuracy of the new equation with three previously reported equations. 2-D and 3-D ultrasound examinations of the thyroid gland were performed in 150 subjects with normal serum thyrotropin (TSH, thyroid-stimulating hormone) and free thyroxine (fT4) levels (63 men and 87 women, age range: 17 to 71 y). In each subject, the volume of both thyroid lobes was measured by 3-D ultrasound. On 2-D ultrasound, the craniocaudal (CC), lateromedial (LM) and anteroposterior (AP) dimensions of the thyroid lobes were measured. The equation was derived by correlating the volume of the thyroid lobes measured with 3-D ultrasound and the product of the three dimensions measured with 2-D ultrasound using linear regression analysis, in 75 subjects without thyroid nodule. The accuracy of thyroid volume estimation of the new equation and the three previously reported equations was evaluated and compared in another 75 subjects (without thyroid nodule, n = 30; with thyroid nodule, n = 45). It is suggested that volume of thyroid lobe may be estimated as: volume of thyroid lobe = 0.38.(CC.LM.AP) + 1.76. Result showed that the new equation (16.9% to 36.1%) had a significantly smaller thyroid volume estimation error than the previously reported equations (20.8% to 54.9%) (p thyroid volume estimation error when thyroid glands with nodules were examined (p thyroid volume equation, 2-D ultrasound can be a useful alternative in thyroid volume measurement when 3-D ultrasound is not available.

  17. A Missed Persistence Property for the Euler Equations, and its Effect on Inviscid Limits

    da Veiga, H Beirao


    We consider the problem of the strong convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations under a Navier slip-type boundary condition to the solution of the Euler equations under the zero flux boundary condition. In spite of the arbitrarily strong convergence results proved in the flat boundary case, see [4], it was shown in reference [5] that the result is false in general, by constructing an explicit family of smooth initial data in the sphere, for which the result fails. Our aim here is to present a more general, simpler and incisive proof. In particular, counterexamples can be displayed in arbitrary, smooth, domains. As in [5], the proof is reduced to the lack of a suitable persistence property for the Euler equations. This negative result is proved by a completely different approach.

  18. Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

    Kulenović MRS


    Full Text Available Abstract We investigate global dynamics of the following systems of difference equations x n + 1 = α 1 + β 1 x n A 1 + y n y n + 1 = γ 2 y n A 2 + B 2 x n + y n , n = 0 , 1 , 2 , … where the parameters α 1, β 1, A 1, γ 2, A 2, B 2 are positive numbers, and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally, in some regions of parameters, we give an explicit formula for the global stable manifold. Mathematics Subject Classification (2000 Primary: 39A10, 39A11 Secondary: 37E99, 37D10

  19. Invariant partial differential equations with two-dimensional exotic centrally extended conformal Galilei symmetry

    Aizawa, N.; Kuznetsova, Z.; Toppan, F.


    Conformal Galilei algebras (CGAs) labeled by d, ℓ (where d is the number of space dimensions and ℓ denotes a spin-ℓ representation w.r.t. the 𝔰𝔩(2) subalgebra) admit two types of central extensions, the ordinary one (for any d and half-integer ℓ) and the exotic central extension which only exists for d = 2 and ℓ ∈ ℕ. For both types of central extensions, invariant second-order partial differential equations (PDEs) with continuous spectrum were constructed by Aizawa et al. [J. Phys. A 46, 405204 (2013)]. It was later proved by Aizawa et al. [J. Math. Phys. 3, 031701 (2015)] that the ordinary central extensions also lead to oscillator-like PDEs with discrete spectrum. We close in this paper the existing gap, constructing a new class of second-order invariant PDEs for the exotic centrally extended CGAs; they admit a discrete and bounded spectrum when applied to a lowest weight representation. These PDEs are markedly different with respect to their ordinary counterparts. The ℓ = 1 case (which is the prototype of this class of extensions, just like the ℓ = /1 2 Schrödinger algebra is the prototype of the ordinary centrally extended CGAs) is analyzed in detail.

  20. One-dimensional tensile constitutive equation cannot be directly generalized to deal with two-dimensional bulging mechanical problems

    SONG; Yuquan(宋玉泉); LIU; Shumei(刘术梅)


    Superplastic forming has been extensively applied to manufacture parts and components with complex shapes or high-precisions. However, superplastic formation is in multi-stress state. In a long time, uniaxial tensile constitutive equation has been directly generalized to deal with multi-stress state. Whether so doing is feasible or not needs to be proved in theory. This paper first summarizes the establishing processes of superplastic tensile and bulging constitutive equation with variable m, and, using the analytical expressions of equivalent stress ? and equivalent strain rateof free bulge based on the fundamentals of continuum medium plastic mechanics, derives the analytical expressions of optimum loading rules for superplastic free bulge. By comparing the quantitative results on typical superplastic alloy ZnAl22, it is shown that one-dimensional tensile constitutive equations cannot be directly generalized to deal with two-dimensional bulging quantitative mechanical problems; only superplastic bulging constitutive equation based on bulging stress state can be used to treat the quantitative mechanical problems of bulge.

  1. Oscillation of solutions to second-order nonlinear differential equations of generalized Euler type

    Asadollah Aghajani


    Full Text Available We are concerned with the oscillatory behavior of the solutions of a generalized Euler differential equation where the nonlinearities satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super linearity are assumed. Some implicit necessary and sufficient conditions and some explicit sufficient conditions are given for all nontrivial solutions of this equation to be oscillatory or nonoscillatory. Also, it is proved that solutions of the equation are all oscillatory or all nonoscillatory and cannot be both.

  2. Analytical solutions of the Schroedinger equation for a two-dimensional exciton in magnetic field of arbitrary strength

    Hoang-Do, Ngoc-Tram; Hoang, Van-Hung; Le, Van-Hoang [Department of Physics, Ho Chi Minh City University of Pedagogy, 280 An Duong Vuong Street, District 5, Ho Chi Minh City (Viet Nam)


    The Feranchuk-Komarov operator method is developed by combining with the Levi-Civita transformation in order to construct analytical solutions of the Schroedinger equation for a two-dimensional exciton in a uniform magnetic field of arbitrary strength. As a result, analytical expressions for the energy of the ground and excited states are obtained with a very high precision of up to four decimal places. Especially, the precision is uniformly stable for the whole range of the magnetic field. This advantage appears due to the consideration of the asymptotic behaviour of the wave-functions in strong magnetic field. The results could be used for various physical analyses and the method used here could also be applied to other atomic systems.

  3. Two-Dimensional Nonlinear Propagation of Ion Acoustic Waves through KPB and KP Equations in Weakly Relativistic Plasmas

    M. G. Hafez


    Full Text Available Two-dimensional three-component plasma system consisting of nonextensive electrons, positrons, and relativistic thermal ions is considered. The well-known Kadomtsev-Petviashvili-Burgers and Kadomtsev-Petviashvili equations are derived to study the basic characteristics of small but finite amplitude ion acoustic waves of the plasmas by using the reductive perturbation method. The influences of positron concentration, electron-positron and ion-electron temperature ratios, strength of electron and positrons nonextensivity, and relativistic streaming factor on the propagation of ion acoustic waves in the plasmas are investigated. It is revealed that the electrostatic compressive and rarefactive ion acoustic waves are obtained for superthermal electrons and positrons, but only compressive ion acoustic waves are found and the potential profiles become steeper in case of subthermal positrons and electrons.

  4. A multi-dimensional kinetic-based upwind solver for the Euler equations

    Eppard, W. M.; Grossman, B.


    A multidimensional kinetic fluctuation-splitting scheme has been developed for the Euler equations. The scheme is based on an N-scheme discretization of the Boltzmann equation at the kinetic level for triangulated Cartesian meshes with a diagonal-adaptive strategy. The resulting Euler scheme is a cell-vertex fluctuation-splitting scheme where fluctuations in the conserved-variable vector Q are obtained as moments of the fluctuation in the Maxwellian velocity distribution function at the kinetic level. Encouraging preliminary results have been obtained for perfect gases on Cartesian meshes with first-order spatial accuracy. The present approach represents an improvement to the well-established dimensionally-split upwind schemes.

  5. An accuracy assessment of Cartesian-mesh approaches for the Euler equations

    Coirier, William J.; Powell, Kenneth G.


    A critical assessment of the accuracy of Cartesian-mesh approaches for steady, transonic solutions of the Euler equations of gas dynamics is made. An exact solution of the Euler equations (Ringleb's flow) is used not only to infer the order of the truncation error of the Cartesian-mesh approaches, but also to compare the magnitude of the discrete error directly to that obtained with a structured mesh approach. Uniformly and adaptively refined solutions using a Cartesian-mesh approach are obtained and compared to each other and to uniformly refined structured mesh results. The effect of cell merging is investigated as well as the use of two different K-exact reconstruction procedures. The solution methodology of the schemes is explained and tabulated results are presented to compare the solution accuracies.

  6. High-Order Spectral Volume Method for 2D Euler Equations

    Wang, Z. J.; Zhang, Laiping; Liu, Yen; Kwak, Dochan (Technical Monitor)


    The Spectral Volume (SV) method is extended to the 2D Euler equations. The focus of this paper is to study the performance of the SV method on multidimensional non-linear systems. Implementation details including total variation diminishing (TVD) and total variation bounded (TVB) limiters are presented. Solutions with both smooth features and discontinuities are utilized to demonstrate the overall capability of the SV method.

  7. Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion

    Kamrani, Minoo; Jamshidi, Nahid


    This work was intended as an attempt to motivate the approximation of quasi linear evolution equations driven by infinite-dimensional fractional Brownian motion with Hurst parameter H >1/2 . The spatial approximation method is based on Galerkin and the temporal approximation is based on implicit Euler scheme. An error bound and the convergence of the numerical method are given. The numerical results show usefulness and accuracy of the method.




    The regular solutions of the isentropic Euler equations with degenerate linear damping for a perfect gas are studied in this paper. And a critical degenerate linear damping coefficient is found, such that if the degenerate linear damping coefficient is larger than it and the gas lies in a compact domain initially, then the regular solution will blow up in finite time; if the degenerate linear damping coefficient is less than it, then undersome hypotheses on the initial data, the regular solution exists globally.

  9. Recursive adjustment approach for the inversion of the Euler-Liouville Equation

    Kirschner, S.; Seitz, F.


    The pole tide Love number (k2) has a direct influence on the numerical forward modeling of the Earth orientation parameters (EOP), polar motion and length of day. This dependency is physically described by the Euler-Liouville equation, which is based on the balance of angular momentum in the Earth system. We use an inversion of Euler-Liouville equation to estimate the pole tide Love number. There are several possible approaches for the implementation of the inversion. Here we concentrate on a recursive adjustment approach. This approach allows the inversion of the Euler-Liouville equation in a computationally efficient way. It is alternative to a Kalman filter approach that will be discussed in the presentation by Seitz et al. in this session. The recursive adjustment approach and its results are analyzed and compared to the results from the Kalman approach. An improvement of the pole tide Love number is possible due to the use of highly precise observed EOP over many decades by geodetic methods. It is shown that the application of the improved pole tide Love number significantly enhances the performance of the forward model. The pole tide Love number effects directly the period and damping of the free rotation of the Earth (Chandler oscillation). The study is carried out in the frame of the German DFG research unit on Earth Rotation and Global Dynamic Processes.

  10. On the use of the incompressibility condition in the Euler and Navier-Stokes equations

    Stubbe, Peter


    The Euler and Navier-Stokes equations both belong to a closed system of three transport equations, describing the particle number density N, the macroscopic velocity v and the temperature T. These sytems are complete, leaving no room for any additional equation. Nonetheless, it is common practice in parts of the literature to replace the thermal equation by the incompressibility condition div v = 0, motivated by the wish to obtain simpler equations. It is shown that this procedure is physically inconsistent in several ways, with the consequence that incompressibility is not a property that can be enforced by an external condition. Incompressible behaviour, if existing, will have to follow self-consistently from the full set of transport equations.



    The authors consider the Euler equations for a compressible fluid in one space dimensionwhen the equation of state of the fluid does not fulfill standard convexity assumptions andviscosity and capillarity effects are taken into account. A typical example of nonconvex con-stitutive equation for fluids is Van der Waals' equation. The first order terms of these partialdifferential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class ofnonconvex equations of state, an existence theorem of traveling waves solutions with arbitrarylarge amplitude is established here. The authors distinguish between classical (compressive) andnonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequali-ties, and are characterized by the so-called kinetic relation, whose properties are investigatedin this paper.

  12. The generalized Euler-Poinsot rigid body equations: explicit elliptic solutions

    Fedorov, Yuri N.; Maciejewski, Andrzej J.; Przybylska, Maria


    The classical Euler-Poinsot case of the rigid body dynamics admits a class of simple but non-trivial integrable generalizations, which modify the Poisson equations describing the motion of the body in space. These generalizations possess first integrals which are polynomial in the angular momenta. We consider the modified Poisson equations as a system of linear equations with elliptic coefficients and show that all the solutions of it are single-valued. By using the vector generalization of the Picard theorem, we derive the solutions explicitly in terms of sigma-functions of the corresponding elliptic curve. The solutions are accompanied by a numerical example. We also compare the generalized Poisson equations with the classical third order Halphen equation.

  13. On reflection symmetry and its application to the Euler-Lagrange equations in fractional mechanics.

    Klimek, Małgorzata


    We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulae are derived for symmetric and anti-symmetric fractional derivatives, both of the Riemann-Liouville and Caputo type. The action dependent on the left-sided Caputo derivatives of orders in the range (1,2) is considered and we derive the Euler-Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler-Lagrange system, we discuss its localization resulting from the subsequent symmetrization of the action.

  14. Weyl-Euler-Lagrange equations on twistor space for tangent structure

    Kasap, Zeki


    Twistor spaces are certain complex three-manifolds, which are associated with special conformal Riemannian geometries on four-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler-Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theory of classical mechanic. This paper aims to introduce Euler-Lagrage partial differential equations (mathematical modeling, the equations of motion according to the time) for the movement of objects on twistor space and also to offer a general solution of differential equation system using the Maple software. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn.

  15. Renormalization group formalism for incompressible Euler equations and the blowup problem

    Mailybaev, Alexei A


    The paper develops the renormalization group (RG) theory for compressible and incompressible inviscid flows, which describes universal scaling of singularities developing in finite (blowup) or infinite time from smooth initial conditions of finite energy. In this theory, the time evolution is substituted by the equivalent evolution given by the RG equations with increasing scaling parameter. Stationary states of the RG equations correspond to self-similar singular solutions. If such a stationary state is an attractor, the corresponding self-similar solution describes universal asymptotic form of a singularity for generic initial conditions. First, we consider the inviscid Burgers equation, where the complete RG analysis is carried out. We prove that the shock formation is described asymptotically by the universal self-similar solution. Then the RG formalism is extended to incompressible Euler equations. Renormalization schemes with single and multiple spatial scales are developed, describing possible asymptot...

  16. General form of the Euler-Poisson-Darboux equation and application of the transmutation method

    Elina L. Shishkina


    Full Text Available In this article, we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler-Poisson-Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter k, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations.

  17. Mechanical problems of superplastic fill-forming bulge solved by one-dimensional tensile and two-dimensional free bulging constitutive equations


    Because of the strong structural sensitivity of superplasticity, the deformation rule must be affected by stress-state. It is necessary to prove whether one-dimensional tensile constitutive equation can be directly generalized to deal with the two-dimensional mechanical problems or not. In this paper, theoretical results of fill-forming bulge have been derived from both one-dimensional tensile and two-dimensional bulging constitutive equation with variable m value. By comparing theoretical analysis and experimental results made on typical superplastic alloy Zn-wt22%Al, it is shown that one-dimensional tensile constitutive equation cannot be directly generalized to deal with two-dimensional mechanical questions. A method to correct deviation between theoretical and experimental results is also proposed.


    高超; 罗时钧; 刘锋


    This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary rectilinear grids. Boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the mean chord line. The method is not restricted to flows with small disturbances since there are no restrictions on the mean angle of attack of the airfoil. The mathematical formulation and the numerical implementation of the wall boundary conditions in a fully implicit time-accurate finite-volume Euler scheme are described. Unsteady transonic flows about an oscillating NACA 0012 airfoil are calculated. Computational results compare well with Euler solutions by the full boundary conditions on a body-fitted curvilinear grid and published experimental data. This study establishes the feasibility for computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in saving computer time and program design since it does not require the generation and implementation of time-dependent body-fitted grids.

  19. Statistical Extremes of Turbulence and a Cascade Generalisation of Euler's Gyroscope Equation

    Tchiguirinskaia, Ioulia; Scherzer, Daniel


    Turbulence refers to a rather well defined hydrodynamical phenomenon uncovered by Reynolds. Nowadays, the word turbulence is used to designate the loss of order in many different geophysical fields and the related fundamental extreme variability of environmental data over a wide range of scales. Classical statistical techniques for estimating the extremes, being largely limited to statistical distributions, do not take into account the mechanisms generating such extreme variability. An alternative approaches to nonlinear variability are based on a fundamental property of the non-linear equations: scale invariance, which means that these equations are formally invariant under given scale transforms. Its specific framework is that of multifractals. In this framework extreme variability builds up scale by scale leading to non-classical statistics. Although multifractals are increasingly understood as a basic framework for handling such variability, there is still a gap between their potential and their actual use. In this presentation we discuss how to dealt with highly theoretical problems of mathematical physics together with a wide range of geophysical applications. We use Euler's gyroscope equation as a basic element in constructing a complex deterministic system that preserves not only the scale symmetry of the Navier-Stokes equations, but some more of their symmetries. Euler's equation has been not only the object of many theoretical investigations of the gyroscope device, but also generalised enough to become the basic equation of fluid mechanics. Therefore, there is no surprise that a cascade generalisation of this equation can be used to characterise the intermittency of turbulence, to better understand the links between the multifractal exponents and the structure of a simplified, but not simplistic, version of the Navier-Stokes equations. In a given way, this approach is similar to that of Lorenz, who studied how the flap of a butterfly wing could generate

  20. A Convergence Result for the Euler-Maruyama Method for a Simple Stochastic Differential Equation with Discontinuous Drift

    Simonsen, Maria; Schiøler, Henrik; Leth, John-Josef


    The Euler-Maruyama method is applied to a simple stochastic differential equation (SDE) with discontinuous drift. Convergence aspects are investigated in the case, where the Euler-Maruyama method is simulated in dyadic points. A strong rate of convergence is presented for the numerical simulations...... and it is shown that the produced sequences converge almost surely. This is an improvement of the general result for SDEs with discontinuous drift, i.e. that the Euler-Maruyama approximations converge in probability to a strong solution of the SDE. A numerical example is presented together with a confidence...


    Wenliang HUANG; Xicheng ZHANG


    Let Xt(x) be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt (x) be the Euler discretization scheme of SDEs with step 2-n . In this note, we prove that for any R>0 and γ∈(0, 1/2), sup t∈[0,1],|x|6R|X nt (x,ω)-Xt (x,ω)|6ξR,γ(ω)2-nγ, n>1, q.e., whereξR,γ(ω) is quasi-everywhere finite.

  2. Self-adjusting entropy-stable scheme for compressible Euler equations

    程晓晗; 聂玉峰; 封建湖; LuoXiao-Yu; 蔡力


    In this work, a self-adjusting entropy-stable scheme is proposed for solving compressible Euler equations. The entropy-stable scheme is constructed by combining the entropy conservative flux with a suitable diffusion operator. The entropy has to be preserved in smooth solutions and be dissipated at shocks. To achieve this, a switch function, based on entropy variables, is employed to make the numerical diffusion term added around discontinuities automatically. The resulting scheme is still entropy-stable. A number of numerical experiments illustrating the robustness and accuracy of the scheme are presented. From these numerical results, we observe a remarkable gain in accuracy.

  3. Field theory and weak Euler-Lagrange equation for classical particle-field systems

    Qin, Hong [PPPL; Burby, Joshua W [PPPL; Davidson, Ronald C [PPPL


    It is commonly believed that energy-momentum conservation is the result of space-time symmetry. However, for classical particle-field systems, e.g., Klimontovich-Maxwell and Klimontovich- Poisson systems, such a connection hasn't been formally established. The difficulty is due to the fact that particles and the electromagnetic fields reside on different manifolds. To establish the connection, the standard Euler-Lagrange equation needs to be generalized to a weak form. Using this technique, energy-momentum conservation laws that are difficult to find otherwise can be systematically derived.

  4. An analysis of flux-split algorithms for Euler's equations with real gases

    Grossman, B.; Walters, R. W.


    An analysis of flux-splitting procedures for the solution of Euler's equations with real gas effects is presented. An alternative real-gas flux-splitting is derived which can easily be implemented into existing codes. This approach, which takes the form of an 'equivalent' gamma representation is not an ad hoc model, but is based on theoretical considerations. Details of this method with the Steger-Warming and Van Leer flux vector splittings and the Roe flux-difference splitting are given. Applications of the method to several high Mach number, high temperature flows are presented for one and two space dimensions.

  5. A rigorous justification of the Euler and Navier-Stokes equations with geometric effects

    Bella, Peter; Lewicka, Marta; Novotny, Antonin


    We derive the 1D isentropic Euler and Navier-Stokes equations describing the motion of a gas through a nozzle of variable cross section as the asymptotic limit of the 3D isentropic Navier-Stokes system in a cylinder, the diameter of which tends to zero. Our method is based on the relative energy inequality satisfied by any weak solution of the 3D Navier-Stokes system and a variant of Korn-Poincare's inequality on thin channels that may be of independent interest.

  6. A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

    Gerritsen, Margot; Olsson, Pelle


    We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.

  7. Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case

    N. N. Nefedov


    Full Text Available Parabolic singularly perturbed problems have been actively studied in recent years in connection with a large number of practical applications: chemical kinetics, synergetics, astrophysics, biology, and so on. In this work a singularly perturbed periodic problem for a parabolic reaction-diffusion equation is studied in the two-dimensional case. The case when there is an internal transition layer under unbalanced nonlinearity is considered. The internal layer is localised near the so called transitional curve. An asymptotic expansion of the solution is constructed and an asymptotics for the transitional curve is determined. The asymptotical expansion consists of a regular part, an interior layer part and a boundary part. In this work we focus on the interior layer part. In order to describe it in the neighborhood of the transition curve the local coordinate system is introduced and the stretched variables are used. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The upper and lower solutions are constructed by sufficiently complicated modification of the asymptotic expansion of the solution. The Lyapunov asymptotical stability of the solution was proved by using the method of contracting barriers. This method is based on the asymptotic comparison principle and uses the upper and lower solutions which are exponentially tending to the solution to the problem. As a result, the solution is locally unique.The article is published in the authors’ wording.


    Bruning, J.; Dobrokhotov, S.Y.; Katsnelson, M.I.; Minenkov, D.S.


    We consider the two-dimensional stationary Schrodinger and Dirac equations in the case of radial symmetry. A radially symmetric potential simulates the tip of a scanning tunneling microscope. We construct semiclassical asymptotic forms for generalized eigenfunctions and study the local density of st

  9. Generalized 2D Euler-Boussinesq equations with a singular velocity

    KC, Durga; Regmi, Dipendra; Tao, Lizheng; Wu, Jiahong


    This paper studies the global (in time) regularity problem concerning a system of equations generalizing the two-dimensional incompressible Boussinesq equations. The velocity here is determined by the vorticity through a more singular relation than the standard Biot-Savart law and involves a Fourier multiplier operator. The temperature equation has a dissipative term given by the fractional Laplacian operator √{-Δ}. We establish the global existence and uniqueness of solutions to the initial-value problem of this generalized Boussinesq equations when the velocity is “double logarithmically” more singular than the one given by the Biot-Savart law. This global regularity result goes beyond the critical case. In addition, we recover a result of Chae, Constantin and Wu [8] when the initial temperature is set to zero.

  10. The Analysis and Construction of Perfectly Matched Layers for the Linearized Euler Equations

    Hesthaven, J. S.


    We present a detailed analysis of a recently proposed perfectly matched layer (PML) method for the absorption of acoustic waves. The split set of equations is shown to be only weakly well-posed, and ill-posed under small low order perturbations. This analysis provides the explanation for the stability problems associated with the split field formulation and illustrates why applying a filter has a stabilizing effect. Utilizing recent results obtained within the context of electromagnetics, we develop strongly well-posed absorbing layers for the linearized Euler equations. The schemes are shown to be perfectly absorbing independent of frequency and angle of incidence of the wave in the case of a non-convecting mean flow. In the general case of a convecting mean flow, a number of techniques is combined to obtain a absorbing layers exhibiting PML-like behavior. The efficacy of the proposed absorbing layers is illustrated though computation of benchmark problems in aero-acoustics.

  11. Elliptical vortex solutions, integrable Ermakov structure, and Lax pair formulation of the compressible Euler equations.

    An, Hongli; Fan, Engui; Zhu, Haixing


    The 2+1-dimensional compressible Euler equations are investigated here. A power-type elliptic vortex ansatz is introduced and thereby reduction obtains to an eight-dimensional nonlinear dynamical system. The latter is shown to have an underlying integral Ermakov-Ray-Reid structure of Hamiltonian type. It is of interest to notice that such an integrable Ermakov structure exists not only in the density representations but also in the velocity components. A class of typical elliptical vortex solutions termed pulsrodons corresponding to warm-core eddy theory is isolated and its behavior is simulated. In addition, a Lax pair formulation is constructed and the connection with stationary nonlinear cubic Schrödinger equations is established.

  12. Persistence of Steady 3D Euler Solutions for 3D Navier-Stokes Equations

    Li, Y Charles


    In the classical plane Couette flow, certain 3D steady solution (the so-called lower branch state) of the Navier-Stokes equations has a nontrivial limit as the Reynolds number approaches infinity \\cite{WGW07}. The limit is a shear of the form ($U(y,z), 0, 0$) in velocity variables. On the other hand, all the shears of this form are solutions of the corresponding 3D Euler equations. This note derives a necessary condition for such a shear to be a limit shear. The condition is $\\int \\Dl U f(U) dy dz = 0$ for any function $f$ satisfying certain boundary condition. Similar conditions are also derived for plane Poiseuille flow and pipe Poiseuille flow, which correspond to similar limit shears as revealed in \\cite{Wal03} \\cite{Vis08}.

  13. A finite element formulation of Euler equations for the solution of steady transonic flows

    Ecer, A.; Akay, H. U.


    The main objective of the considered investigation is related to the development of a relaxation scheme for the analysis of inviscid, rotational, transonic flow problems. To formulate the equations of motion for inviscid flows in a fixed coordinate system, an Eulerian type variational principle is required. The derivation of an Eulerian variational principle which is employed in the finite element formulation is discussed. The presented numerical method describes the mathematical formulation and the application of a numerical process for the direct solution of steady Euler equations. The development of the procedure as an extension of existing potential flow formulations provides the applicability of previous procedures, e.g., proper application of the artificial viscosity for supersonic elements, and the accurate modeling of the shock.

  14. Approximate Solutions of Nonlinear Fractional Kolmogorov-Petrovskii-Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method

    宋丽娜; 王维国


    By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

  15. Approximate Solutions of Nonlinear Fractional Kolmogorov—Petrovskii—Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method

    Song, Li-Na; Wang, Wei-Guo


    By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

  16. Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow

    Frisch, Uriel


    Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to G\\"ottingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler flow and can be extended to compressible flow, in particular to cosmological dark matter. Hankel showed that Cauchy's formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation the...

  17. New Methods for Two-Dimensional Schr\\"odinger Equation SUSY-separation of Variables and Shape Invariance

    Cannata, F; Nishnianidze, D N


    Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one - $SUSY-$ separation of variables - is based on the intertwining relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges allowing for separation of variables. 2)The second one is a generalization of shape invariance. While in one dimension shape invariance allows to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been yet explored. Here we provide a formal framework in HSUSY QM for two-dimensional quantum mechanical systems for which shape invariance holds. Given the knowledge of one eigenvalue and eigenfunction, shape invariance allows to construct a chain of new eigenfunctions and eigenvalues. These methods are applied to a two-dimensional quantum system, and partial explicit solvability is achieved in the sense that only part of the spectrum is found analyt...

  18. Exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength

    Hoang-Do, Ngoc-Tram [Department of Physics, Ho Chi Minh City University of Pedagogy 280, An Duong Vuong Street, District 5, Ho Chi Minh City (Viet Nam); Pham, Dang-Lan [Institute for Computational Science and Technology, Quang Trung Software Town, District 12, Ho Chi Minh City (Viet Nam); Le, Van-Hoang, E-mail: [Department of Physics, Ho Chi Minh City University of Pedagogy 280, An Duong Vuong Street, District 5, Ho Chi Minh City (Viet Nam)


    Exact numerical solutions of the Schrödinger equation for a two-dimensional exciton in a constant magnetic field of arbitrary strength are obtained for not only the ground state but also high excited states. Toward this goal, the operator method is developed by combining with the Levi-Civita transformation which transforms the problem under investigation into that of a two-dimensional anharmonic oscillator. This development of the non-perturbation method is significant because it can be applied to other problems of two-dimensional atomic systems. The obtained energies and wave functions set a new record for their precision of up to 20 decimal places. Analyzing the obtained data we also find an interesting result that exact analytical solutions exist at some values of magnetic field intensity.

  19. Stability and non-relativistic limits of rarefaction wave to the 1-D piston problem for the relativistic Euler equations

    Ding, Min; Li, Yachun


    We study the 1-D piston problem for the relativistic Euler equations under the assumption that the total variations of both the initial data and the velocity of the piston are sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Meanwhile, we consider the convergence of the entropy solutions to the corresponding entropy solutions of the classical non-relativistic Euler equations as the light speed c→ +∞.

  20. Quasi-periodic non-stationary solutions of 3D Euler equations for incompressible flow

    Ershkov, Sergey V


    A novel derivation of non-stationary solutions of 3D Euler equations for incompressible inviscid flow is considered here. Such a solution is the product of 2 separated parts: - one consisting of the spatial component and the other being related to the time dependent part. Spatial part of a solution could be determined if we substitute such a solution to the equations of motion (equation of momentum) with the requirement of scale-similarity in regard to the proper component of spatial velocity. So, the time-dependent part of equations of momentum should depend on the time-parameter only. The main result, which should be outlined, is that the governing (time-dependent) ODE-system consist of 2 Riccati-type equations in regard to each other, which has no solution in general case. But we obtain conditions when each component of time-dependent part is proved to be determined by the proper elliptical integral in regard to the time-parameter t, which is a generalization of the class of inverse periodic functions.

  1. Newton-Euler Dynamic Equations of Motion for a Multi-body Spacecraft

    Stoneking, Eric


    The Magnetospheric MultiScale (MMS) mission employs a formation of spinning spacecraft with several flexible appendages and thruster-based control. To understand the complex dynamic interaction of thruster actuation, appendage motion, and spin dynamics, each spacecraft is modeled as a tree of rigid bodies connected by spherical or gimballed joints. The method presented facilitates assembling by inspection the exact, nonlinear dynamic equations of motion for a multibody spacecraft suitable for solution by numerical integration. The building block equations are derived by applying Newton's and Euler's equations of motion to an "element" consisting of two bodies and one joint (spherical and gimballed joints are considered separately). Patterns in the "mass" and L'force" matrices guide assembly by inspection of a general N-body tree-topology system. Straightforward linear algebra operations are employed to eliminate extraneous constraint equations, resulting in a minimum-dimension system of equations to solve. This method thus combines a straightforward, easily-extendable, easily-mechanized formulation with an efficient computer implementation.

  2. Conservative discontinuous Galerkin discretizations of the 2D incompressible Euler equation

    Waelbroeck, Francois; Michoski, Craig; Bernard, Tess


    Discontinuous Galerkin (DG) methods provide local high-order adaptive numerical schemes for the solution of convection-diffusion problems. They combine the advantages of finite element and finite volume methods. In particular, DG methods automatically ensure the conservation of all first-order invariants provided that single-valued fluxes are prescribed at inter-element boundaries. For the 2D incompressible Euler equation, this implies that the discretized fluxes globally obey Gauss' and Stokes' laws exactly, and that they conserve total vorticity. Liu and Shu have shown that combining a continuous Galerkin (CG) solution of Poisson's equation with a central DG flux for the convection term leads to an algorithm that conserves the principal two quadratic invariants, namely the energy and enstrophy. Here, we present a discretization that applies the DG method to Poisson's equation as well as to the vorticity equation while maintaining conservation of the quadratic invariants. Using a DG algorithm for Poisson's equation can be advantageous when solving problems with mixed Dirichlet-Neuman boundary conditions such as for the injection of fluid through a slit (Bickley jet) or during compact toroid injection for tokamak startup.

  3. Stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable duffing oscillator and bifurcation of moment equation

    Zhang Guangjun [School of Aerospace, Xi' an Jiao Tong University, Xi' an (China) and School of Life and Science and Technology, Xi' an Jiao Tong University, Xi' an (China) and School of Science, Air Force Engineering University, Xi' an (China)], E-mail:; Xu Jianxue [School of Aerospace, Xi' an Jiao Tong University, Xi' an (China)], E-mail:; Wang Jue [School of Life and Science and Technology, Xi' an Jiao Tong University, Xi' an (China); Yue Zhifeng; Zou Hailin [School of Aerospace, Xi' an Jiao Tong University, Xi' an (China)


    In this paper stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable Duffing oscillator is analyzed by moment method. This kind of novel transition refers to the one among three potential well on two sides of bifurcation point of original system at the presence of internal noise. Several conclusions are drawn. First, the semi-analytical result of stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable Duffing oscillator can be obtained, and the semi-analytical result is qualitatively compatible with the one of Monte Carlo simulation. Second, a bifurcation of double-branch fixed point curves occurs in the moment equations with noise intensity as their bifurcation parameter. Third, the bifurcation of moment equations corresponds to stochastic resonance of original system. Finally, the mechanism of stochastic resonance is presented from another viewpoint through analyzing the energy transfer induced by the bifurcation of moment equation.

  4. Exact solutions of a two-dimensional cubic–quintic discrete nonlinear Schrödinger equation

    Khare, Avinash; Rasmussen, Kim Ø; Samuelsen, Mogens Rugholm


    We show that a two-dimensional generalized cubic–quintic Ablowitz–Ladik lattice admits periodic solutions that can be expressed in analytical form. The framework for the stability analysis of these solutions is developed and applied to reveal the intricate stability behavior of this nonlinear sys...




    A numerical treatment for the prediction of cavitating flows is presented and assessed.The algorithm uses the preconditioned multiphase Euler equations with appropriate mass transfer terms.A central difference finite volume scheme with suitable dissipation terms to account for density jumps across the cavity interface is shown to yield an effective method for solving the multiphase Euler equations.The Euler equations are utilized herein for the cavitation modeling, because some certain characteristics of cavitating flows can be obtained using the solution of this system of equations with relative low computational effort.In addition, the Euler equations are appropriate for the assessment of the numerical method used, because of the sensitivity of the solution to the numerical instabilities.For this reason, a sensitivity study is conducted to evaluate the effects of various parameters, such as numerical dissipation coefficients and grid size, on the accuracy and performance of the solution.The computations are performed for steady cavitating flows around the NACA 0012 and NACA 66 (MOD) hydrofoils and also an axisymmetric hemispherical fore-body under different conditions and the results are compared with the available numerical and experimental data.The solution procedure presented is shown to be accurate and efficient for predicting steady sheet- and super-cavitation for 2D/axisymmetric geometries.

  6. Well-balanced nodal discontinuous Galerkin method for Euler equations with gravity

    Chandrashekar, Praveen


    We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

  7. Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry.

    Cheung, Ka Luen; Wong, Sen


    The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the N-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form c(t)|x| (α-1) x + b(t)(x/|x|) for any value of α ≠ 1 or any positive integer N ≠ 1. Then, we show that blowup phenomenon occurs when α = N = 1 and [Formula: see text]. As a corollary, the blowup properties of solutions with velocity of the form [Formula: see text] are obtained. Our analysis includes both the isentropic case (γ > 1) and the isothermal case (γ = 1).

  8. Analytical study of sandwich structures using Euler-Bernoulli beam equation

    Xue, Hui; Khawaja, H.


    This paper presents an analytical study of sandwich structures. In this study, the Euler-Bernoulli beam equation is solved analytically for a four-point bending problem. Appropriate initial and boundary conditions are specified to enclose the problem. In addition, the balance coefficient is calculated and the Rule of Mixtures is applied. The focus of this study is to determine the effective material properties and geometric features such as the moment of inertia of a sandwich beam. The effective parameters help in the development of a generic analytical correlation for complex sandwich structures from the perspective of four-point bending calculations. The main outcomes of these analytical calculations are the lateral displacements and longitudinal stresses for each particular material in the sandwich structure.

  9. Well-posedness of compressible Euler equations in a physical vacuum

    Jang, Juhi


    An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only few mathematical results available near vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local-in-time well-posedness of three-dimensional compressible Euler equations for polytropic gases with physical vacuum by considering the problem as a free boundary problem.

  10. A Dynamically Adaptive Arbitrary Lagrangian-Eulerian Method for Solution of the Euler Equations

    Anderson, R W; Elliott, N S; Pember, R B


    A new method that combines staggered grid arbitrary Lagrangian-Eulerian (ALE) techniques with structured local adaptive mesh refinement (AMR) has been developed for solution of the Euler equations. The novel components of the methods are driven by the need to reconcile traditional AMR techniques with the staggered variables and moving, deforming meshes associated with Lagrange based ALE schemes. We develop interlevel solution transfer operators and interlevel boundary conditions first in the case of purely Lagrangian hydrodynamics, and then extend these ideas into an ALE method by developing adaptive extensions of elliptic mesh relaxation techniques. Conservation properties of the method are analyzed, and a series of test problem calculations are presented which demonstrate the utility and efficiency of the method.

  11. Strong Ill-posedness of the incompressible Euler equation in borderline Sobolev spaces

    Bourgain, Jean


    For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem. In this paper we consider the physical dimensions $d=2,3$ and show that if we perturb any given smooth initial data in $H^{s_c}$ norm, then the corresponding solution can have infinite $H^{s_c}$ norm instantaneously at $t>0$. The constructed solutions are unique and even $C^{\\infty}$-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.

  12. A Multigrid Block LU-SGS Algorithm for Euler Equations on Unstructured Grids

    Ruo Li; Xin Wang; Weibo Zhao


    We propose an efficient and robust algorithm to solve the steady Euler equations on unstructured grids. The new algorithm is a Newton-iteration method in which each iteration step is a linear multigrid method using block lower-upper symmetric Gauss-Seidel (LU-SGS) iteration as its smoother. To regularize the Jacobian matrix of Newton-iteration, we adopted a local residual dependent regularization as the replace ment of the standard time-stepping relaxation technique based on the local CFL number. The proposed method can be extended to high order approximations and three spatial dimensions in a nature way. The solver was tested on a sequence of benchmark prob lems on both quasi-uniform and local adaptive meshes. The numerical results illustrated the efficiency and robustness of our algorithm.


    张映辉; 吴国春


    We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

  14. The Degasperis-Procesi equation as a non-metric Euler equation

    Escher, Joachim


    In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of $b$-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\\'echet space of all smooth functions on the circle.

  15. Geometric field theory and weak Euler-Lagrange equation for classical relativistic particle-field systems

    Fan, Peifeng; Liu, Jian; Xiang, Nong; Yu, Zhi


    A manifestly covariant, or geometric, field theory for relativistic classical particle-field system is developed. The connection between space-time symmetry and energy-momentum conservation laws for the system is established geometrically without splitting the space and time coordinates, i.e., space-time is treated as one identity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that particles and field reside on different manifold. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of electromagnetic fields and also a functional of particles' world-lines. The other difficulty associated with the geometric setting is due to the mass-shell condition. The standard Euler-Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell condition is imposed. For the particle-field system, the geometric EL equation is further generalized into a w...

  16. Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes: The constraint equations

    Brauer, Uwe; Karp, Lavi

    This paper deals with the construction of initial data for the coupled Einstein-Euler system. We consider the condition where the energy density might vanish or tend to zero at infinity, and where the pressure is a fractional power of the energy density. In order to achieve our goals we use a type of weighted Sobolev space of fractional order. The common Lichnerowicz-York scaling method (Choquet-Bruhat and York, 1980 [9]; Cantor, 1979 [7]) for solving the constraint equations cannot be applied here directly. The basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources. This problem has been addressed, although in a different context, by Dain and Nagy (2002) [11]. We show that if the matter variables are restricted to a certain region, then the Einstein constraint equations have a unique solution in the weighted Sobolev spaces of fractional order. The regularity depends upon the fractional power of the equation of state.

  17. A Preconditioning Method for Shape Optimization Governed by the Euler Equations

    Arian, Eyal; Vatsa, Veer N.


    We consider a classical aerodynamic shape optimization problem subject to the compressible Euler flow equations. The gradient of the cost functional with respect to the shape variables is derived with the adjoint method at the continuous level. The Hessian (second order derivative of the cost functional with respect to the shape variables) is approximated also at the continuous level, as first introduced by Arian and Ta'asan (1996). The approximation of the Hessian is used to approximate the Newton step which is essential to accelerate the numerical solution of the optimization problem. The design space is discretized in the maximum dimension, i.e., the location of each point on the intersection of the computational mesh with the airfoil is taken to be an independent design variable. We give numerical examples for 86 design variables in two different flow speeds and achieve an order of magnitude reduction in the cost functional at a computational effort of a full solution of the analysis partial differential equation (PDE).

  18. A stochastic Galerkin method for the Euler equations with Roe variable transformation

    Pettersson, Per


    The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy. © 2013 Elsevier Inc.

  19. An Eulerian-Lagrangian Form for the Euler Equations in Sobolev Spaces

    Pooley, Benjamin C.; Robinson, James C.


    In 2000 Constantin showed that the incompressible Euler equations can be written in an "Eulerian-Lagrangian" form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Hölder spaces {C^{1,μ}}. We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data in H s for {n ≥ 2} and {s > n/2+1}, a unique local-in-time solution exists on the n-torus that is continuous into H s and C 1 into H s-1. These solutions automatically have C 1 trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian-Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory.

  20. A Computational Investigation of the Finite-Time Blow-Up of the 3D Incompressible Euler Equations Based on the Voigt Regularization

    Larios, Adam; Titi, Edriss S; Wingate, Beth


    We report the results of a computational investigation of two recently proved blow-up criteria for the 3D incompressible Euler equations. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations. The latter are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $\\alpha>0$. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be sufficient criteria for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.

  1. Improving Solution of Euler Equations by a Gas-Kinetic BGK Method

    Liu Ya; Gao Chao; F. Liu


    Aim. The well known JST(Jameson-Schmidt-Turkel) scheme requires the use of a dissipation term. We propose using gas-kinetic BGK (Bhatnagar-Gross-Krook) method, which is based on the more fundamental Boltzmann equation, in order to obviate the use of dissipation term and obtain, we believe, an improved solution. Section 1 deals essentially with three things: (1) as analytical solution of molecular probability density function at the ceil interface has been obtained by the Bohzmann equation with BGK model, we can compute the flux term by integrating the density function in the phase space; eqs. (8) and (11) require careful attention; (2) the integrations can be expressed as the moments of Maxwellian distribution with different limits according to the analytical solution; eqs. (9) and (10) require careful attention; (3) the discrete equation by finite volume method can be solved using the time marching method. Computations are performed by the BGK method for the Sod's shock tube problem and a two-dimensional shock reflection problem. The results are compared with those of the conventional JST scheme in Figs. 1 and 2. The BGK method provides better resolution of shock waves and other features of the flow fields.

  2. Scale-selective dissipation in energy-conserving finite element schemes for two-dimensional turbulence

    Natale, Andrea


    We analyse the multiscale properties of energy-conserving upwind-stabilised finite element discretisations of the two-dimensional incompressible Euler equations. We focus our attention on two particular methods: the Lie derivative discretisation introduced in Natale and Cotter (2016a) and the SUPG discretisation of the vorticity advection equation. Such discretisations provide control on enstrophy by modelling different types of scale interactions. We quantify the performance of the schemes in reproducing the non-local energy backscatter that characterises two-dimensional turbulent flows.

  3. Beam stabilization in the two-dimensional nonlinear Schrodinger equation with an attractive potential by beam splitting and radiation

    leMesurier, B.J.; Christiansen, Peter Leth; Gaididei, Yuri Borisovich


    The effect of attractive linear potentials on self-focusing in-waves modeled by a nonlinear Schrodinger equation is considered. It is shown that the attractive potential can prevent both singular collapse and dispersion that are generic in the cubic Schrodinger equation in the critical dimension 2...

  4. Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

    Qingxue Huang


    Full Text Available In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

  5. Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

    Abdi, Daniel S.; Giraldo, Francis X.


    A unified approach for the numerical solution of the 3D hyperbolic Euler equations using high order methods, namely continuous Galerkin (CG) and discontinuous Galerkin (DG) methods, is presented. First, we examine how classical CG that uses a global storage scheme can be constructed within the DG framework using constraint imposition techniques commonly used in the finite element literature. Then, we implement and test a simplified version in the Non-hydrostatic Unified Model of the Atmosphere (NUMA) for the case of explicit time integration and a diagonal mass matrix. Constructing CG within the DG framework allows CG to benefit from the desirable properties of DG such as, easier hp-refinement, better stability etc. Moreover, this representation allows for regional mixing of CG and DG depending on the flow regime in an area. The different flavors of CG and DG in the unified implementation are then tested for accuracy and performance using a suite of benchmark problems representative of cloud-resolving scale, meso-scale and global-scale atmospheric dynamics. The value of our unified approach is that we are able to show how to carry both CG and DG methods within the same code and also offer a simple recipe for modifying an existing CG code to DG and vice versa.

  6. Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation

    Bispen, Georgij; Lukáčová-Medvid'ová, Mária; Yelash, Leonid


    In this paper we will present and analyze a new class of the IMEX finite volume schemes for the Euler equations with a gravity source term. We will in particular concentrate on a singular limit of weakly compressible flows when the Mach number M ≪ 1. In order to efficiently resolve slow dynamics we split the whole nonlinear system in a stiff linear part governing the acoustic and gravity waves and a non-stiff nonlinear part that models nonlinear advection effects. For time discretization we use a special class of the so-called globally stiffly accurate IMEX schemes and approximate the stiff linear operator implicitly and the non-stiff nonlinear operator explicitly. For spatial discretization the finite volume approximation is used with the central and Rusanov/Lax-Friedrichs numerical fluxes for the linear and nonlinear subsystem, respectively. In the case of a constant background potential temperature we prove theoretically that the method is asymptotically consistent and asymptotically stable uniformly with respect to small Mach number. We also analyze experimentally convergence rates in the singular limit when the Mach number tends to zero.

  7. A multigrid method for steady Euler equations on unstructured adaptive grids

    Riemslagh, Kris; Dick, Erik


    A flux-difference splitting type algorithm is formulated for the steady Euler equations on unstructured grids. The polynomial flux-difference splitting technique is used. A vertex-centered finite volume method is employed on a triangular mesh. The multigrid method is in defect-correction form. A relaxation procedure with a first order accurate inner iteration and a second-order correction performed only on the finest grid, is used. A multi-stage Jacobi relaxation method is employed as a smoother. Since the grid is unstructured a Jacobi type is chosen. The multi-staging is necessary to provide sufficient smoothing properties. The domain is discretized using a Delaunay triangular mesh generator. Three grids with more or less uniform distribution of nodes but with different resolution are generated by successive refinement of the coarsest grid. Nodes of coarser grids appear in the finer grids. The multigrid method is started on these grids. As soon as the residual drops below a threshold value, an adaptive refinement is started. The solution on the adaptively refined grid is accelerated by a multigrid procedure. The coarser multigrid grids are generated by successive coarsening through point removement. The adaption cycle is repeated a few times. Results are given for the transonic flow over a NACA-0012 airfoil.

  8. Cartesian Mesh Linearized Euler Equations Solver for Aeroacoustic Problems around Full Aircraft

    Yuma Fukushima


    Full Text Available The linearized Euler equations (LEEs solver for aeroacoustic problems has been developed on block-structured Cartesian mesh to address complex geometry. Taking advantage of the benefits of Cartesian mesh, we employ high-order schemes for spatial derivatives and for time integration. On the other hand, the difficulty of accommodating curved wall boundaries is addressed by the immersed boundary method. The resulting LEEs solver is robust to complex geometry and numerically efficient in a parallel environment. The accuracy and effectiveness of the present solver are validated by one-dimensional and three-dimensional test cases. Acoustic scattering around a sphere and noise propagation from the JT15D nacelle are computed. The results show good agreement with analytical, computational, and experimental results. Finally, noise propagation around fuselage-wing-nacelle configurations is computed as a practical example. The results show that the sound pressure level below the over-the-wing nacelle (OWN configuration is much lower than that of the conventional DLR-F6 aircraft configuration due to the shielding effect of the OWN configuration.

  9. Entropy and weak solutions in the thermal model for the compressible Euler equations

    Ran, Zheng


    Among the existing models for compressible fluids, the one by Kataoka and Tsutahara (KT model, Phys. Rev. E 69, 056702, 2004) has a simple and rigorous theoretical background. The drawback of this KT model is that it can cause numerical instability if the local Mach number exceeds 1. The precise mechanism of this instability has not yet been clarified. In this paper, we derive entropy functions whose local equilibria are suitable to recover the Euler-like equations in the framework of the lattice Boltzmann method for the KT model. Numerical examples are also given, which are consistent with the above theoretical arguments, and show that the entropy condition is not fully guaranteed in KT model. The negative entropy may be the inherent cause for the non-physical oscillations in the vicinity of the shock. In contrast to these Karlin's microscopic entropy approach, the corresponding subsidiary entropy condition in the LBM calculation could also be deduced explicitly from the macroscopic version, which provides s...

  10. The lifespan of 3D radial solutions to the non-isentropic relativistic Euler equations

    Wei, Changhua


    This paper investigates the lower bound of the lifespan of three-dimensional spherically symmetric solutions to the non-isentropic relativistic Euler equations, when the initial data are prescribed as a small perturbation with compact support to a constant state. Based on the structure of the hyperbolic system, we show the almost global existence of the smooth solutions to Eulerian flows (polytropic gases and generalized Chaplygin gases) with genuinely nonlinear characteristics. While for the Eulerian flows (Chaplygin gas and stiff matter) with mild linearly degenerate characteristics, we show the global existence of the radial solutions, moreover, for the non-strictly hyperbolic system (pressureless perfect fluid) satisfying the mild linearly degenerate condition, we prove the blowup phenomenon of the radial solutions and show that the lifespan of the solutions is of order O(ɛ ^{-1}), where ɛ denotes the width of the perturbation. This work can be seen as a complement of our work (Lei and Wei in Math Ann 367:1363-1401, 2017) for relativistic Chaplygin gas and can also be seen as a generalization of the classical Eulerian fluids (Godin in Arch Ration Mech Anal 177:497-511, 2005, J Math Pures Appl 87:91-117, 2007) to the relativistic Eulerian fluids.

  11. Nonlinear acoustic propagation in two-dimensional ducts

    Nayfeh, A. H.; Tsai, M.-S.


    The method of multiple scales is used to obtain a second-order uniformly valid expansion for the nonlinear acoustic wave propagation in a two-dimensional duct whose walls are treated with a nonlinear acoustic material. The wave propagation in the duct is characterized by the unsteady nonlinear Euler equations. The results show that nonlinear effects tend to flatten and broaden the absorption versus frequency curve, in qualitative agreement with the experimental observations. Moreover, the effect of the gas nonlinearity increases with increasing sound frequency, whereas the effect of the material nonlinearity decreases with increasing sound frequency.

  12. Adaptive finite-volume WENO schemes on dynamically redistributed grids for compressible Euler equations

    Pathak, Harshavardhana S.; Shukla, Ratnesh K.


    A high-order adaptive finite-volume method is presented for simulating inviscid compressible flows on time-dependent redistributed grids. The method achieves dynamic adaptation through a combination of time-dependent mesh node clustering in regions characterized by strong solution gradients and an optimal selection of the order of accuracy and the associated reconstruction stencil in a conservative finite-volume framework. This combined approach maximizes spatial resolution in discontinuous regions that require low-order approximations for oscillation-free shock capturing. Over smooth regions, high-order discretization through finite-volume WENO schemes minimizes numerical dissipation and provides excellent resolution of intricate flow features. The method including the moving mesh equations and the compressible flow solver is formulated entirely on a transformed time-independent computational domain discretized using a simple uniform Cartesian mesh. Approximations for the metric terms that enforce discrete geometric conservation law while preserving the fourth-order accuracy of the two-point Gaussian quadrature rule are developed. Spurious Cartesian grid induced shock instabilities such as carbuncles that feature in a local one-dimensional contact capturing treatment along the cell face normals are effectively eliminated through upwind flux calculation using a rotated Hartex-Lax-van Leer contact resolving (HLLC) approximate Riemann solver for the Euler equations in generalized coordinates. Numerical experiments with the fifth and ninth-order WENO reconstructions at the two-point Gaussian quadrature nodes, over a range of challenging test cases, indicate that the redistributed mesh effectively adapts to the dynamic flow gradients thereby improving the solution accuracy substantially even when the initial starting mesh is non-adaptive. The high adaptivity combined with the fifth and especially the ninth-order WENO reconstruction allows remarkably sharp capture of


    莫则尧; 符尚武


    Two dimensional three temperatures energy equation is a kind of very impor-tant partial differential equation. In general, we discrete such equation with full implicit nine points stencil on Lagrange structured grid and generate a non-linear sparse algebraic equation including nine diagonal lines. This paper will discuss the iterative solver for such non-linear equations. We linearize the equations by fixing the coefficient matrix, and iteratively solve the linearized algebraic equation with Krylov subspace iterative method. We have applied the iterative method presented in this paper to the code Lared-Ⅰ for numerical simulation of two dimensional threetemperatures radial fluid dynamics, and have obtained efficient results.

  14. On a relation of pseudoanalytic function theory to the two-dimensional stationary Schroedinger equation and Taylor series in formal powers for its solutions

    Kravchenko, Vladislav V [Seccion de Posgrado e Investigacion, Escuela Superior de IngenierIa Mecanica y Electrica, Instituto Politecnico Nacional, C.P.07738 Mexico DF (Mexico)


    We consider the real stationary two-dimensional Schroedinger equation. With the aid of any of its particular solutions, we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schroedinger equation and the imaginary parts are solutions of an associated Schroedinger equation with a potential having the form of a potential obtained after the Darboux transformation. Using Bers' theory of Taylor series for pseudoanalytic functions, we obtain a locally complete system of solutions of the original Schroedinger equation which can be constructed explicitly for an ample class of Schroedinger equations. For example it is possible when the potential is a function of one Cartesian, spherical, parabolic or elliptic variable. We give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the Schroedinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation.

  15. Ising and Bloch domain walls in a two-dimensional parametrically driven Ginzburg-Landau equation model with nonlinearity management

    Gaididei, Yu. B.; Christiansen, Peter Leth


    We study a parametrically driven Ginzburg-Landau equation model with nonlinear management. The system is made of laterally coupled long active waveguides placed along a circumference. Stationary solutions of three kinds are found: periodic Ising states and two types of Bloch states, staggered...... and unstaggered. The stability of these states is investigated analytically and numerically. The nonlinear dynamics of the Bloch states are described by a complex Ginzburg-Landau equation with linear and nonlinear parametric driving. The switching between the staggered and unstaggered Bloch states under...

  16. Boundary and Interface Conditions for High Order Finite Difference Methods Applied to the Euler and Navier-Strokes Equations

    Nordstrom, Jan; Carpenter, Mark H.


    Boundary and interface conditions for high order finite difference methods applied to the constant coefficient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.

  17. Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation

    Konopelchenko, B [Dipartimento di Fisica, Universita di Lecce and Sezione INFN, 73100 Lecce (Italy); Alonso, L MartInez [Departamento de Fisica Teorica II, Universidad Complutense, E28040 Madrid (Spain); Medina, E [Departamento de Matematicas, Universidad de Cadiz, E11510 Puerto Real, Cadiz (Spain)


    It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock-type singularities are presented.

  18. CABARET scheme for the numerical solution of aeroacoustics problems: Generalization to linearized one-dimensional Euler equations

    Goloviznin, V. M.; Karabasov, S. A.; Kozubskaya, T. K.; Maksimov, N. V.


    A generalization of the CABARET finite difference scheme is proposed for linearized one-dimensional Euler equations based on the characteristic decomposition into local Riemann invariants. The new method is compared with several central finite difference schemes that are widely used in computational aeroacoustics. Numerical results for the propagation of an acoustic wave in a homogeneous field and the refraction of this wave through a contact discontinuity obtained on a strongly nonuniform grid are presented.

  19. Stochastic Liouville equations for hydrogen-bonding fluctuations and their signatures in two-dimensional vibrational spectroscopy of water

    Jansen, TL; Hayashi, T; Zhuang, W; Mukamel, S


    The effects of hydrogen-bond forming and breaking kinetics on the linear and coherent third-order infrared spectra of the OH stretch of HOD in D2O are described by Markovian, not necessarily Gaussian, fluctuations and simulated using the stochastic Liouville equations. Slow (0.5 ps) fluctuations are

  20. A new numerical method for solving two-dimensional variable-order anomalous sub-diffusion equation

    Jiang Wei


    Full Text Available The novelty and innovativeness of this paper are the combination of reproducing kernel theory and spline, this leads to a new simple but effective numerical method for solving variable-order anomalous sub-diffusion equation successfully. This combination overcomes the weaknesses of piecewise polynomials that can not be used to solve differential equations directly because of lack of the smoothness. Moreover, new bases of reproducing kernel spaces are constructed. On the other hand, the existence of any ε-approximate solution is proved and an effective method for obtaining the ε-approximate solution is established. A numerical example is given to show the accuracy and effectiveness of theoretical results.

  1. Solution of Two-dimensional Parabolic Equation Subject to Non-local Boundary Conditions Using Homotopy Perturbation Method

    Puskar Raj SHARMA


    Full Text Available Aim of the paper is to investigate solution of twodimensional linear parabolic partial differential equation with non-local boundary conditions using Homotopy Perturbation Method (HPM. This method is not only reliable in obtaining solution of such problems in series form with high accuracy but it also guarantees considerable saving of the calculation volume and time as compared to other methods. The application of the method has been illustrated through an example

  2. Asymptotic Behaviour of Solutions to the Navier-stokes Equations of a Two-dimensional Compressible Flow

    Ying-hui ZHANG; Zhong TAN


    In this paper,we are concerned with the asymptotic behaviour of a weak solution to the NavierStokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p(ρ) =a(ρ)logd(ρ) for large (ρ).Here d > 2,a > 0.We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity.Using properties of this function,we can prove the strong convergence of the density to its limit state.The behaviour of the velocity field and kinetic energy is also briefly discussed.

  3. Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations

    Guermond, Jean-Luc; Popov, Bojan


    This paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the Riemann solution for the Euler system of gas dynamics with the co-volume equation of state. The novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy. The convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co-volume equation of state and the heat capacity ratio γ in the range (1 , 5 / 3 ].

  4. Equation of state calculations for two-dimensional dust coulomb crystal at near zero temperature by molecular dynamics simulations

    Djouder, M., E-mail:; Kermoun, F.; Mitiche, M. D.; Lamrous, O. [Laboratoire de Physique et Chimie Quantique, Université Mouloud Mammeri Tizi-Ouzou, BP 17 RP, 15000 Tizi-Ouzou (Algeria)


    Dust particles observed in universe as well as in laboratory and technological plasma devices are still under investigation. At low temperature, these particles are strongly negatively charged and are able to form a 2D or 3D coulomb crystal. In this work, our aim was to check the ideal gas law validity for a 2D single-layer dust crystal recently reported in the literature. For this purpose, we have simulated, using the molecular dynamics method, its thermodynamic properties for different values of dust particles number and confinement parameters. The obtained results have allowed us to invalidate the ideal gas behaviour and to propose an effective equation of state which assumes a near zero dust temperature. Furthermore, the value of the calculated sound velocity was found to be in a good agreement with experimental data published elsewhere.

  5. Extracting trajectory equations of classical periodic orbits from the quantum eigenmodes in two-dimensional integrable billiards

    Hsieh, Y. H.; Yu, Y. T.; Tuan, P. H.; Tung, J. C.; Huang, K. F.; Chen, Y. F.


    The trajectory equations for classical periodic orbits in the equilateral-triangular and circular billiards are systematically extracted from quantum stationary coherent states. The relationship between the phase factors of quantum stationary coherent states and the initial positions of classical periodic orbits is analytically derived. In addition, the stationary coherent states with noncoprime parametric numbers are shown to correspond to the multiple periodic orbits, which cannot be explicable in the one-particle picture. The stationary coherent states are further verified to be linked to the resonant modes that are generally observed in the experimental wave system excited by a localized and unidirectional source. The excellent agreement between the resonant modes and the stationary coherent states not only manifests the importance of classical features in experimental systems but also paves the way to manipulate the mesoscopic wave functions localized on the periodic orbits for applications.

  6. Local membrane length conservation in two-dimensional vesicle simulation using a multicomponent lattice Boltzmann equation method.

    Halliday, I; Lishchuk, S V; Spencer, T J; Pontrelli, G; Evans, P C


    We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013)10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.

  7. On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions

    Settle, Sean O.


    The primary aim of this paper is to answer the question, What are the highest-order five- or nine-point compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one- and two-dimensional Poisson equation on uniform, quasi-uniform, and nonuniform face-to-face hyperrectangular grids and directly prove the existence or nonexistence of their highest-order local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the one-dimensional problem, the derivation using the three-point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high-order local accuracy. However, for the two-dimensional problem, the derivation using the corresponding five-point stencil on uniform and quasi-uniform grids yields a scheme with at most second-order local accuracy, and on nonuniform grids yields at most first-order local accuracy. When expanding the five-point stencil to the nine-point stencil, the derivation using the nine-point stencil on uniform grids yields at most sixth-order local accuracy, but on quasi- and nonuniform grids yields at most fourth- and third-order local accuracy, respectively. © 2013 Society for Industrial and Applied Mathematics.

  8. High-order compact MacCormack scheme for two-dimensional compressible and non-hydrostatic equations of the atmosphere

    JavanNezhad, R.; Meshkatee, A. H.; Ghader, S.; Ahmadi-Givi, F.


    This study is devoted to application of the fourth-order compact MacCormack scheme to spatial differencing of the conservative form of two-dimensional and non-hydrostatic equation of a dry atmosphere. To advance the solution in time a four-stage Runge-Kutta method is used. To perform the simulations, two test cases including evolution of a warm bubble and a cold bubble in a neutral atmosphere with open and rigid boundaries are employed. In addition, the second-order MacCormack and the standard fourth-order compact MacCormack schemes are used to perform the simulations. Qualitative and quantitative assessment of the numerical results for different test cases exhibit the superiority of the fourth-order compact MacCormack scheme on the second-order method.

  9. Computing a numerical solution of two dimensional non-linear Schrödinger equation on complexly shaped domains by RBF based differential quadrature method

    Golbabai, Ahmad; Nikpour, Ahmad


    In this paper, two-dimensional Schrödinger equations are solved by differential quadrature method. Key point in this method is the determination of the weight coefficients for approximation of spatial derivatives. Multiquadric (MQ) radial basis function is applied as test functions to compute these weight coefficients. Unlike traditional DQ methods, which were originally defined on meshes of node points, the RBFDQ method requires no mesh-connectivity information and allows straightforward implementation in an unstructured nodes. Moreover, the calculation of coefficients using MQ function includes a shape parameter c. A new variable shape parameter is introduced and its effect on the accuracy and stability of the method is studied. We perform an analysis for the dispersion error and different internal parameters of the algorithm are studied in order to examine the behavior of this error. Numerical examples show that MQDQ method can efficiently approximate problems in complexly shaped domains.

  10. Determination of scale-invariant equations of state without fitting parameters: application to the two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition.

    Desbuquois, Rémi; Yefsah, Tarik; Chomaz, Lauriane; Weitenberg, Christof; Corman, Laura; Nascimbène, Sylvain; Dalibard, Jean


    We present a general "fit-free" method for measuring the equation of state (EoS) of a scale-invariant gas. This method, which is inspired from the procedure introduced by Ku et al. [Science 335, 563 (2012)] for the unitary three-dimensional Fermi gas, provides a general formalism which can be readily applied to any quantum gas in a known trapping potential, in the frame of the local density approximation. We implement this method on a weakly interacting two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition and determine its EoS with unprecedented accuracy in the critical region. Our measurements provide an important experimental benchmark for classical-field approaches which are believed to accurately describe quantum systems in the weakly interacting but nonperturbative regime.

  11. A finite difference technique for solving a time strain separable K-BKZ constitutive equation for two-dimensional moving free surface flows

    Tomé, M. F.; Bertoco, J.; Oishi, C. M.; Araujo, M. S. B.; Cruz, D.; Pinho, F. T.; Vynnycky, M.


    This work is concerned with the numerical solution of the K-BKZ integral constitutive equation for two-dimensional time-dependent free surface flows. The numerical method proposed herein is a finite difference technique for simulating flows possessing moving surfaces that can interact with solid walls. The main characteristics of the methodology employed are: the momentum and mass conservation equations are solved by an implicit method; the pressure boundary condition on the free surface is implicitly coupled with the Poisson equation for obtaining the pressure field from mass conservation; a novel scheme for defining the past times t‧ is employed; the Finger tensor is calculated by the deformation fields method and is advanced in time by a second-order Runge-Kutta method. This new technique is verified by solving shear and uniaxial elongational flows. Furthermore, an analytic solution for fully developed channel flow is obtained that is employed in the verification and assessment of convergence with mesh refinement of the numerical solution. For free surface flows, the assessment of convergence with mesh refinement relies on a jet impinging on a rigid surface and a comparison of the simulation of a extrudate swell problem studied by Mitsoulis (2010) [44] was performed. Finally, the new code is used to investigate in detail the jet buckling phenomenon of K-BKZ fluids.

  12. Asymptotic Behavior of Global Solution for Nonlinear Generalized Euler-Possion-Darboux Equation%一类非线性广义Euler-Poisson-Darboux方程整体解的渐近性质

    梁保松; 陈振


    J. L Lions and W. A. Stranss [1] have proved the existence of a global solution of the initial boundary value problem for nonlinear generalized Euler-Possion-Darboux equation. In this paper we are going to investigate the asymptotic behavior of the global solution by a difference inequality.

  13. An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations

    Pan, Liang; Xu, Kun; Li, Qibing; Li, Jiequan


    For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the second-order gas-kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around a cell interface. With the adoption of time derivative of the flux function, a two-stage Lax-Wendroff-type (L-W for short) time stepping method has been recently proposed in the design of a fourth-order time accurate method for inviscid flow [21]. In this paper, based on the same time-stepping method and the second-order GKS flux function [42], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier-Stokes (NS) equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [24], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme. In terms of the computational cost, a two-dimensional third-order GKS flux function takes about six times of the computational time of a second-order GKS flux function. However, a fifth-order WENO reconstruction may take more than ten times of the computational cost of a second-order GKS flux function. Therefore, it is fully legitimate to develop a two-stage fourth order time accurate method (two reconstruction) instead of standard four stage fourth-order Runge-Kutta method (four reconstruction). Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. In the current computational fluid dynamics (CFD) research, it is still a difficult problem to extend the higher-order Euler solver to the NS one due to the change of governing equations from hyperbolic to parabolic type and the initial interface discontinuity. This problem remains distinctively for the hypersonic viscous and heat conducting flow. The GKS is based on the kinetic equation with the hyperbolic transport and the relaxation source term. The time-dependent GKS flux function

  14. Adaptation of the Euler-Lagrange equation for studying one-dimensional motions in a constant force

    Dias, Clenilda F; Silva, Gislene M; Santos, Creuza A S; Barros, Pedro; Carvalho-Santos, Vagson L


    In this work we have shown that the Euler-Lagrange equation (ELE) can be simplified for one-dimensional motions. By using the partial derivative operators definition, we have proposed two operators, here called \\textit{mean delta operators}, which may be used to solve the ELE in a simplest way. We have applied this simplification to solve three known mechanical problems: a free fall body, the Atwood's machine and the inclinated plan. The proposed simplification may be used for introducing the lagrangian formalism for classical mechanics in introductory physics students, e.g., high school or undergraduate students in the beginning of engineering, mathematics and/or physics courses.

  15. A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations

    Shakib, Farzin; Hughes, Thomas J. R.; Johan, Zdenek


    A space-time element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms and boundary conditions. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. Numerical results are presented to demonstrate the performance of the method.

  16. A Novel Method for the Numerical Solution of the Navier-Stokes Equations in Two-Dimensional Flow Using a Pressure Poisson Equation

    Messaris, G. T.; Papastavrou, C. A.; Loukopoulos, V. C.; Karahalios, G. T.


    A new finite-difference method is presented for the numerical solution of the Navier-Stokes equations of motion of a viscous incompressible fluid in two (or three) dimensions and in the primitive-variable formulation. Introducing two auxiliary functions of the coordinate system and considering the form of the initial equation on lines passing through the nodal point (x0, y0) and parallel to the coordinate axes, we can separate it into two parts that are finally reduced to ordinary differential equations, one for each dimension. The final system of linear equations in n-unknowns is solved by an iterative technique and the method converges rapidly giving satisfactory results. For the pressure variable we consider a pressure Poisson equation with suitable Neumann boundary conditions. Numerical results, confirming the accuracy of the proposed method, are presented for configurations of interest, like Poiseuille flow and the flow between two parallel plates with step under the presence of a pressure gradient.

  17. Dynamical equations for the vector potential and the velocity potential in incompressible irrotational Euler flows: a refined Bernoulli theorem.

    Ohkitani, Koji


    We consider incompressible Euler flows in terms of the stream function in two dimensions and the vector potential in three dimensions. We pay special attention to the case with singular distributions of the vorticity, e.g., point vortices in two dimensions. An explicit equation governing the velocity potentials is derived in two steps. (i) Starting from the equation for the stream function [Ohkitani, Nonlinearity 21, T255 (2009)NONLE50951-771510.1088/0951-7715/21/12/T02], which is valid for smooth flows as well, we derive an equation for the complex velocity potential. (ii) Taking a real part of this equation, we find a dynamical equation for the velocity potential, which may be regarded as a refinement of Bernoulli theorem. In three-dimensional incompressible flows, we first derive dynamical equations for the vector potentials which are valid for smooth fields and then recast them in hypercomplex form. The equation for the velocity potential is identified as its real part and is valid, for example, flows with vortex layers. As an application, the Kelvin-Helmholtz problem has been worked out on the basis the current formalism. A connection to the Navier-Stokes regularity problem is addressed as a physical application of the equations for the vector potentials for smooth fields.

  18. Hamilton's Equations with Euler Parameters for Rigid Body Dynamics Modeling. Chapter 3

    Shivarama, Ravishankar; Fahrenthold, Eric P.


    A combination of Euler parameter kinematics and Hamiltonian mechanics provides a rigid body dynamics model well suited for use in strongly nonlinear problems involving arbitrarily large rotations. The model is unconstrained, free of singularities, includes a general potential energy function and a minimum set of momentum variables, and takes an explicit state space form convenient for numerical implementation. The general formulation may be specialized to address particular applications, as illustrated in several three dimensional example problems.

  19. Implementation of duffing system based on Euler equations%Duffing系统的欧拉实现方法研究

    芮国胜; 史特; 张洋


    在研究Duffing系统的基础上,为了利用Duffing系统进行弱信号检测,必须实现一种可靠的Duffing系统模型.对3种基于欧拉方程的模型实现方式进行了对比,通过分析欧拉方程的几何意义,并结合DSP Builder的特点,提出了一种易于工程实现的改进欧拉形式.通过对四种算法进行仿真实验得出,运用不同的状态模型建立Duffing系统会改变振子相变的临界策动力幅值,但不会对系统固有存在的状态产生影响.%In order to use the Duffing system to detect weak signal; a reliable Duffing system model must be achieved. Three realization modes of model based on Euler equations are compared. By analyzing the geometric meaning of the Euler equations and combining the characteristics of DSP Builder, an improved Euler form which is easy to realize in engineering is proposed on the basis of the related research on Duffing system. The simulation experiments for four algorithms indicate that the establishment of Buffing system with different state models can change the critical driving force amplitude of the oscillator phase transition, but can not affect the inherent status of the system.

  20. An efficient high-order compact scheme for the unsteady compressible Euler and Navier-Stokes equations

    Lerat, A.


    Residual-Based Compact (RBC) schemes approximate the 3-D compressible Euler equations with a 5th- or 7th-order accuracy on a 5 × 5 × 5-point stencil and capture shocks pretty well without correction. For unsteady flows however, they require a costly algebra to extract the time-derivative occurring at several places in the scheme. A new high-order time formulation has been recently proposed [13] for simplifying the RBC schemes and increasing their temporal accuracy. The present paper goes much further in this direction and deeply reconsiders the method. An avatar of the RBC schemes is presented that greatly reduces the computing time and the memory requirements while keeping the same type of successful numerical dissipation. Two and three-dimensional linear stability are analyzed and the method is extended to the 3-D compressible Navier-Stokes equations. The new compact scheme is validated for several unsteady problems in two and three dimension. In particular, an accurate DNS at moderate cost is presented for the evolution of the Taylor-Green Vortex at Reynolds 1600 and Prandtl 0.71. The effects of the mesh size and of the accuracy order in the approximation of Euler and viscous terms are discussed.

  1. Initial and Boundary Value Problems for Two-Dimensional Non-hydrostatic Boussinesq Equations%二维非静力Boussinesq方程组的初边值问题

    沈春; 孙梅娜


    Based on the theory of stratification, the well-posedness of the initial and boundary value problems for the system of two-dimensional non-hydrostatic Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for some representative initial and boundary value problems. Several special cases were discussed.

  2. H2-Stabilization of the Isothermal Euler Equations:a Lyapunov Function Approach

    Martin GUGAT; Günter LEUGERING; Simona TAMASOIU; Ke WANG


    The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this end,an explicit Lyapunov function as a weighted and squared H2-norm of a small perturbation of the stationary solution is constructed.The authors show that by a suitable choice of the boundary feedback conditions,the H2-exponential stability of the stationary solution follows.Due to this fact,the system is stabilized over an infinite time interval.Furthermore,exponential estimates for the C1-norm are derived.

  3. 随机微分方程的Euler数值解法%Euler methods for numerical solution of stochastic ordinary differential equations



    Based on the well-established numercal methods for the deterministic ordinary differential equations the Duler method is applied to a scalar autonomous stochastic ordinary differential equation and three Euler numerical schemes are given:explicit scheme;semi-implicit sheme and implicit scheme.One type of stability of the Euler method ,T-stability ,is considered.Numerical results of a linear test equation show that the Euler method for solving SODEs is meaningful.%基于常微分方程(ODEs)的Euler数值解法,提出了求解一类随机常微分方程(ODEs)的3种Euler格式:显Euler格式,半隐Euler格式和隐Euler格式.讨论了3种Euler格式的T-稳定条件,并给出了部分数值实验结果.

  4. Large scale spatio-temporal behaviour in surface growth. Scaling and dynamics of slow height variations in generalized two-dimensional Kuramoto-Sivashinsky equations

    Juknevičius, Vaidas; Ruseckas, Julius; Armaitis, Jogundas


    This paper presents new findings concerning the dynamics of the slow height variations in surfaces produced by the two-dimensional isotropic Kuramoto-Sivashinsky equation with an additional nonlinear term. In addition to the disordered cellular patterns of specific size evident at small scales, slow height variations of scale-free character become increasingly evident when the system size is increased. This paper focuses on the parameter range in which the kinetic roughening with eventual saturation in surface roughness and coarseness is obtained, and the statistical and dynamical properties of surfaces in the long-time stationary regime are investigated. The resulting long-range scaling properties of the saturated surface roughness consistent with the power-law shape of the surface spectrum at small wave numbers are obtained in a wider parameter range than previously reported. The temporal properties of these long-range height variations are investigated by analysing the time series of surface roughness fluctuations. The resulting power-spectral densities can be expressed as a generalized Lorentzian whose cut-off frequency varies with system size. The dependence of this lower cut-off frequency on the smallest wave number connects spatial and temporal properties and gives new insight into the surface evolution on large scales.

  5. Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation.

    Asllanaj, Fatmir; Fumeron, Sebastien


    Optical tomography is a medical imaging technique based on light propagation in the near infrared (NIR) part of the spectrum. We present a new way of predicting the short-pulsed NIR light propagation using a time-dependent two-dimensional-global radiative transfer equation in an absorbing and strongly anisotropically scattering medium. A cell-vertex finite-volume method is proposed for the discretization of the spatial domain. The closure relation based on the exponential scheme and linear interpolations was applied for the first time in the context of time-dependent radiative heat transfer problems. Details are given about the application of the original method on unstructured triangular meshes. The angular space (4πSr) is uniformly subdivided into discrete directions and a finite-differences discretization of the time domain is used. Numerical simulations for media with physical properties analogous to healthy and metastatic human liver subjected to a collimated short-pulsed NIR light are presented and discussed. As expected, discrepancies between the two kinds of tissues were found. In particular, the level of light flux was found to be weaker (inside the medium and at boundaries) in the healthy medium than in the metastatic one.

  6. A New Flux Splitting Scheme Based on Toro-Vazquez and HLL Schemes for the Euler Equations

    Pascalin Tiam Kapen


    Full Text Available This paper presents a new flux splitting scheme for the Euler equations. The proposed scheme termed TV-HLL is obtained by following the Toro-Vazquez splitting (Toro and Vázquez-Cendón, 2012 and using the HLL algorithm with modified wave speeds for the pressure system. Here, the intercell velocity for the advection system is taken as the arithmetic mean. The resulting scheme is more accurate when compared to the Toro-Vazquez schemes and also enjoys the property of recognition of contact discontinuities and shear waves. Accuracy, efficiency, and other essential features of the proposed scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. The accuracy of the scheme is shown in 1D test cases designed by Toro.

  7. Euler-Lagrange equations for holomorphic structures on twistorial generalized Kähler manifolds

    Zeki Kasap


    showing motion modeling partial di¤erential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the Maple software. Additionally, of the implicit solution of the equations to be drawn the graph.

  8. Hadamard States and Two-dimensional Gravity

    Salehi, H


    We have used a two-dimensional analog of the Hadamard state-condition to study the local constraints on the two-point function of a linear quantum field conformally coupled to a two-dimensional gravitational background. We develop a dynamical model in which the determination of the state of the quantum field is essentially related to the determination of a conformal frame. A particular conformal frame is then introduced in which a two-dimensional gravitational equation is established.

  9. Surface boundary conditions for the numerical solution of the Euler equations

    Dadone, A.; Grossman, B.


    We consider the implementation of boundary conditions at solid walls in inviscid Euler solutions by upwind, finite-volume methods. We review some current methods for the implementation of surface boundary conditions and examine their behavior for the problem of an oblique shock reflecting off a planar surface. We show the importance of characteristic boundary conditions for this problem and introduce a method of applying the classical flux-difference splitting of Roe as a characteristic boundary condition. Consideration of the equivalent problem of the intersection of two (equal and opposite) oblique shocks was very illuminating on the role of surface boundary conditions for an inviscid flow and led to the introduction of two new boundary-condition procedures, denoted as the symmetry technique and the curvature-corrected symmetry technique. Examples of the effects of the various surface boundary conditions considered are presented for the supersonic blunt body problem and the subcritical compressible flow over a circular cylinder. Dramatic advantages of the curvature-corrected symmetry technique over the other methods are shown, with regard to numerical entropy generation, total pressure loss, drag and grid convergence.

  10. Maxwell's equations as a special case of deformation of a solid lattice in Euler's coordinates

    Gremaud, G


    It is shown that the set of equations known as Maxwell's equations perfectly describe two very different systems: (1) the usual electromagnetic phenomena in vacuum or in the matter and (2) the deformation of isotropic solid lattices, containing topological defects as dislocations and disclinations, in the case of constant and homogenous expansion. The analogy between these two physical systems is complete, as it is not restricted to one of the two Maxwell's equation couples in the vacuum, but generalized to the two equation couples as well as to the diverse phenomena of dielectric polarization and magnetization of matter, just as to the electrical charges and the electrical currents. The eulerian approach of the solid lattice developed here includes Maxwell's equations as a special case, since it stems from a tensor theory, which is reduced to a vector one by contraction on the tensor indices. Considering the tensor aspect of the eulerian solid lattice deformation theory, the analogy can be extended to other ...

  11. 若干包含Euler函数φ(n)的方程%Some Equations Involving Euler Function φ( n )

    孙翠芳; 程智


    For any positive integer n, let φ(n) be Euler function and Ω(n) denote the total number of prime factors of re, we studied the solutions of three kinds of equations n -φ(n) =2Ω(n) , n-φ(φ(n)) =2Ω(n) and φ(n -φ(n)) =2Ω(n) using the methods of number theory, obtaining all the positive integer solutions of these kinds of equations.%设n为任意正整数,φ(n)是Euler函数,Ω(n)表示n的素因数个数.利用数论中的理论和方法,研究三类方程n-φ(n)=2Ω(n),n-φ(φ(n))=2Ω(n)和φ(n-φ(n))=2Ω(n)的可解性问题,获得了这三类方程的所有正整数解.

  12. Integration by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell

    Ibragimov, Nail H. [Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona (Sweden); Ibragimov, Ranis N., E-mail: [Department of Mathematics, College of Science, Mathematics and Technology, University of Texas at Brownsville, TX 78520 (United States)


    We study the nonlinear incompressible non-viscous fluid flows within a thin rotating atmospheric shell that serve as a simple mathematical description of an atmospheric circulation caused by the temperature difference between the equator and the poles. The model is also superimposed by a particular stationary flow which, under the assumption of no friction and a distribution of temperature dependent only upon latitude, models the zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. Owing to the Coriolis effects, the resulting achievable meteorological flows correspond to the asymptotical stable flows that are being translated along the equatorial plane. The exact solutions in terms of elementary functions are found by using Lie group methods. -- Highlights: → This article provides new exact solutions of the Euler and Navier-Stokes equations. → The exact solutions are written in terms of elementary functions. → The exact solutions were obtained by Lie group analysis. → A wider class of exact solutions is contained in the obtained Lie algebra.

  13. High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves

    Christiansen, Torben Robert Bilgrav; Bingham, Harry B.; Engsig-Karup, Allan Peter


    is discretized using arbitrary-order finite difference schemes on a staggered grid with one optional stretching in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order Runge-Kutta scheme. Mass conservation is satisfied...

  14. An adaptive-gridding solution method for the 2D unsteady Euler equations

    Wackers, J.


    Adaptive grid refinement is a technique to speed up the numerical solution of partial differential equations by starting these calculations on a coarse basic grid and refining this grid only there where the solution requires this, e.g. in areas with large gradients. This technique has already been u

  15. Is classical mechanics based on Newton's laws or Eulers analytical equations?



    Full Text Available In an example I illustrate how my picture of physics is enriched due to my frequent conversations with Reinhard Folk. The subject is: Who wrote down the basic equations of motion of classical mechanics for the first time? (To be sure, it was not Newton.

  16. Is classical mechanics based on Newton's laws or Eulers analytical equations?

    Iro, H


    In an example I illustrate how my picture of physics is enriched due to my frequent conversations with Reinhard Folk. The subject is: Who wrote down the basic equations of motion of classical mechanics for the first time? (To be sure, it was not Newton.)

  17. High-Order Non-Reflecting Boundary Conditions for the Linearized Euler Equations


    Bernoulli, Chebyshev, Fibonacci , Hermite, Legendre, Laguerre, Spread, Touchard, Rook, Orthogonal, Secondary, Sheffer sequence , Sturm se- quence, and...Pante Stǎnicǎ here at NPS for his discovery concerning the polynomial sequence discussed on p. 108. I am of course deeply indebted to the Air Force for...equation and deriving Padé approximations thereto in a sequence of ever-more-accurate boundary conditions. Smith [101] took a simplistic, albeit

  18. Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painleve VI equation

    Levin, A; Zotov, A


    In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on $R$-matrix description which provides Lax pairs in terms of quantum and classical $R$-matrices. First, we prove that for relativistic (and non-relativistic) tops such Lax pairs with spectral parameter follow from the associative Yang-Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of ${\\mathbb Z}_2$ reduced elliptic top and verify that the latter constraints are fulfilled in this case. The construction of matrix extensions is naturally generalized to the monodromy preserving equation. In this way we get matrix extensions of the Painlev\\'e VI equation and its multidimensional analogues written in the form of non-autonomous elliptic tops. Finally, it is mentioned that the matrix valued variables can be replaced by elements of noncommut...

  19. Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painlevé VI equation

    Levin, A.; Olshanetsky, M.; Zotov, A.


    In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops, such Lax pairs with spectral parameters follow from the associative Yang-Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of the {{{Z}}}2 reduced elliptic top and verify that the latter constraints are fulfilled in this case. The construction of matrix extensions is naturally generalized to the monodromy preserving equation. In this way we get matrix extensions of the Painlevé VI equation and its multidimensional analogues written in the form of non-autonomous elliptic tops. Finally, it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra. At the end of the paper we also describe special elliptic Gaudin models which can be considered as matrix extensions of the ({{{Z}}}2 reduced) elliptic top.

  20. Utilization of Euler-Lagrange Equations in Circuits with Memory Elements

    Z. Biolek


    Full Text Available It is well known that the equation of motion of a system can be set up using the Lagrangian and the dissipation function, which describe the conservative and dissipative parts of the system. However, this procedure, consisting in a systematic differentiation of the above state functions, cannot be used for circuits containing simultaneously conventional nonlinear elements such as the resistor, capacitor, and inductor, and their nonlinear memory versions – the memristor, memcapacitor, and meminductor. The paper provides a general solution to this problem and demonstrates it on the example of modeling Josephson’s junction.

  1. Testing the functional equations of a high-degree Euler product

    Farmer, David W; Schmidt, Ralf


    We study the L-functions associated to Siegel modular forms (equivalently, automorphic representations of ${\\rm GSp}(4,\\mathbb{A}_{\\mathbb{Q}})$) both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform representation theoretic calculations to cast the Langlands L-function in classical terms. We develop a precise notion of what it means to test a conjectured functional equation for an L-function, and we apply this to the degree 10 adjoint L-function associated to a Siegel modular form.

  2. Hamiltonian discontinuous Galerkin FEM for linear, stratified (in)compressible Euler equations: internal gravity waves

    van Oers, Alexander M.; Maas, Leo R. M.; Bokhove, Onno


    The linear equations governing internal gravity waves in a stratified ideal fluid possess a Hamiltonian structure. A discontinuous Galerkin finite element method has been developed in which this Hamiltonian structure is discretized, resulting in conservation of discrete analogs of phase space and energy. This required (i) the discretization of the Hamiltonian structure using alternating flux functions and symplectic time integration, (ii) the discretization of a divergence-free velocity field using Dirac's theory of constraints and (iii) the handling of large-scale computational demands due to the 3-dimensional nature of internal gravity waves and, in confined, symmetry-breaking fluid domains, possibly its narrow zones of attraction.

  3. Exponential Attractor for the Derivative Two dimensional Ginaburg-Landau Equation in Banach Spaces%二维广义Ginzburg-Landau方程在Banach空间的指数吸引子

    黄健; 戴正德


    在本文中,我们在Banach空间考虑二维广义Ginzburg-Landau方程的指数吸引子,且得到其分形维度估计.%In this paper, we consider the exponential attractor for the derivative two - dimensional Ginzburg - Landau equation in Banach space Xαp and also obtain the estimation of the fractal dimension.

  4. Hydrodynamics for a model of a confined quasi-two-dimensional granular gas.

    Brey, J Javier; Buzón, V; Maynar, P; García de Soria, M I


    The hydrodynamic equations for a model of a confined quasi-two-dimensional gas of smooth inelastic hard spheres are derived from the Boltzmann equation for the model, using a generalization of the Chapman-Enskog method. The heat and momentum fluxes are calculated to Navier-Stokes order, and the associated transport coefficients are explicitly determined as functions of the coefficient of normal restitution and the velocity parameter involved in the definition of the model. Also an Euler transport term contributing to the energy transport equation is considered. This term arises from the gradient expansion of the rate of change of the temperature due to the inelasticity of collisions, and it vanishes for elastic systems. The hydrodynamic equations are particularized for the relevant case of a system in the homogeneous steady state. The relationship with previous works is analyzed.

  5. Two-dimensional calculus

    Osserman, Robert


    The basic component of several-variable calculus, two-dimensional calculus is vital to mastery of the broader field. This extensive treatment of the subject offers the advantage of a thorough integration of linear algebra and materials, which aids readers in the development of geometric intuition. An introductory chapter presents background information on vectors in the plane, plane curves, and functions of two variables. Subsequent chapters address differentiation, transformations, and integration. Each chapter concludes with problem sets, and answers to selected exercises appear at the end o

  6. Two dimensional vernier

    Juday, Richard D. (Inventor)


    A two-dimensional vernier scale is disclosed utilizing a cartesian grid on one plate member with a polar grid on an overlying transparent plate member. The polar grid has multiple concentric circles at a fractional spacing of the spacing of the cartesian grid lines. By locating the center of the polar grid on a location on the cartesian grid, interpolation can be made of both the X and Y fractional relationship to the cartesian grid by noting which circles coincide with a cartesian grid line for the X and Y direction.

  7. On a complex differential Riccati equation

    Khmelnytskaya, Kira V; Kravchenko, Vladislav V [Department of Mathematics, CINVESTAV del IPN, Unidad Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 Mexico (Mexico)], E-mail:


    We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schroedinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation such as the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical 'one-dimensional' results, we discuss new features of the considered equation including an analogue of the Cauchy integral theorem.

  8. Calculation of the rotor-fuselage interaction using the resolution of Euler equations; Calcul de l'interaction rotor-fuselage par resolution des equations d'Euler

    Bettschart, N.


    This paper presents the recent developments made at ONERA for the prediction of helicopter rotor-fuselage aerodynamic interactions. An actuator disk model has been implemented in the FLU3M code. Applications on isolated rotors as well as complete helicopter are performed using the Euler option of the code. The numerical results are compared with theoretical and experimental results. (author)

  9. Calculation of the rotor-fuselage interaction using the resolution of Euler equations; Calcul de l'interaction rotor-fuselage par resolution des equations d'Euler

    Bettschart, N.


    This paper presents the recent developments made at ONERA for the prediction of helicopter rotor-fuselage aerodynamic interactions. An actuator disk model has been implemented in the FLU3M code. Applications on isolated rotors as well as complete helicopter are performed using the Euler option of the code. The numerical results are compared with theoretical and experimental results. (author)

  10. Two-dimensional capillary origami

    Brubaker, N.D., E-mail:; Lega, J., E-mail:


    We describe a global approach to the problem of capillary origami that captures all unfolded equilibrium configurations in the two-dimensional setting where the drop is not required to fully wet the flexible plate. We provide bifurcation diagrams showing the level of encapsulation of each equilibrium configuration as a function of the volume of liquid that it contains, as well as plots representing the energy of each equilibrium branch. These diagrams indicate at what volume level the liquid drop ceases to be attached to the endpoints of the plate, which depends on the value of the contact angle. As in the case of pinned contact points, three different parameter regimes are identified, one of which predicts instantaneous encapsulation for small initial volumes of liquid. - Highlights: • Full solution set of the two-dimensional capillary origami problem. • Fluid does not necessarily wet the entire plate. • Global energy approach provides exact differential equations satisfied by minimizers. • Bifurcation diagrams highlight three different regimes. • Conditions for spontaneous encapsulation are identified.

  11. Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional navier-stokes equations and Newton's laws of motion

    Carling; Williams; Bowtell


    Anguilliform swimming has been investigated by using a computational model combining the dynamics of both the creature's movement and the two-dimensional fluid flow of the surrounding water. The model creature is self-propelled; it follows a path determined by the forces acting upon it, as generated by its prescribed changing shape. The numerical solution has been obtained by applying coordinate transformations and then using finite difference methods. Results are presented showing the flow around the creature as it accelerates from rest in an enclosed tank. The kinematics and dynamics associated with the creature's centre of mass are also shown. For a particular set of body shape parameters, the final mean swimming speed is found to be 0.77 times the speed of the backward-travelling wave. The corresponding movement amplitude envelope is shown. The magnitude of oscillation in the net forward force has been shown to be approximately twice that in the lateral force. The importance of allowing for acceleration and deceleration of the creature's body (rather than imposing a constant swimming speed) has been demonstrated. The calculations of rotational movement of the body and the associated moment of forces about the centre of mass have also been included in the model. The important role of viscous forces along and around the creature's body and in the growth and dissolution of the vortex structures has been illustrated.

  12. Two-dimensional optical spectroscopy

    Cho, Minhaeng


    Discusses the principles and applications of two-dimensional vibrational and optical spectroscopy techniques. This book provides an account of basic theory required for an understanding of two-dimensional vibrational and electronic spectroscopy.

  13. Uniformly accurate Particle-in-Cell method for the long time solution of the two-dimensional Vlasov-Poisson equation with uniform strong magnetic field

    Crouseilles, Nicolas; Lemou, Mohammed; Méhats, Florian; Zhao, Xiaofei


    In this work, we focus on the numerical resolution of the four dimensional phase space Vlasov-Poisson system subject to a uniform strong external magnetic field. To do so, we consider a Particle-in-Cell based method, for which the characteristics are reformulated by means of the two-scale formalism, which is well-adapted to handle highly-oscillatory equations. Then, a numerical scheme is derived for the two-scale equations. The so-obtained scheme enjoys a uniform accuracy property, meaning that its accuracy does not depend on the small parameter. Several numerical results illustrate the capabilities of the method.

  14. A parametrization of two-dimensional turbulence based on a maximum entropy production principle with a local conservation of energy

    Chavanis, Pierre-Henri


    In the context of two-dimensional (2D) turbulence, we apply the maximum entropy production principle (MEPP) by enforcing a local conservation of energy. This leads to an equation for the vorticity distribution that conserves all the Casimirs, the energy, and that increases monotonically the mixing entropy ($H$-theorem). Furthermore, the equation for the coarse-grained vorticity dissipates monotonically all the generalized enstrophies. These equations may provide a parametrization of 2D turbulence. They do not generally relax towards the maximum entropy state. The vorticity current vanishes for any steady state of the 2D Euler equation. Interestingly, the equation for the coarse-grained vorticity obtained from the MEPP turns out to coincide, after some algebraic manipulations, with the one obtained with the anticipated vorticity method. This shows a connection between these two approaches when the conservation of energy is treated locally. Furthermore, the newly derived equation, which incorporates a diffusion...

  15. Beam stabilization in the two-dimensional nonlinear Schrödinger equation with an attractive potential by beam splitting and radiation.

    leMesurier, Brenton John; Christiansen, Peter Leth; Gaididei, Yuri B; Rasmussen, Jens Juul


    The effect of attractive linear potentials on self-focusing in-waves modeled by a nonlinear Schrödinger equation is considered. It is shown that the attractive potential can prevent both singular collapse and dispersion that are generic in the cubic Schrödinger equation in the critical dimension 2 and can lead to a stable oscillating beam. This is observed to involve a splitting of the beam into an inner part that is oscillatory and of subcritical power and an outer dispersing part. An analysis is given in terms of the rate competition between the linear and nonlinear focusing effects, radiation losses, and known stable periodic behavior of certain solutions in the presence of attractive potentials.

  16. An Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations. Ph.D. Thesis - Michigan Univ.

    Coirier, William John


    A Cartesian, cell-based scheme for solving the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal 'cut' cells are created. The geometry of the cut cells is computed using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded, with a limited linear reconstruction of the primitive variables used to provide input states to an approximate Riemann solver for computing the fluxes between neighboring cells. A multi-stage time-stepping scheme is used to reach a steady-state solution. Validation of the Euler solver with benchmark numerical and exact solutions is presented. An assessment of the accuracy of the approach is made by uniform and adaptive grid refinements for a steady, transonic, exact solution to the Euler equations. The error of the approach is directly compared to a structured solver formulation. A non smooth flow is also assessed for grid convergence, comparing uniform and adaptively refined results. Several formulations of the viscous terms are assessed analytically, both for accuracy and positivity. The two best formulations are used to compute adaptively refined solutions of the Navier-Stokes equations. These solutions are compared to each other, to experimental results and/or theory for a series of low and moderate Reynolds numbers flow fields. The most suitable viscous discretization is demonstrated for geometrically-complicated internal flows. For flows at high Reynolds numbers, both an altered grid-generation procedure and a

  17. Finite differences numerical method for two-dimensional superlattice Boltzmann transport equation and case comparison of CPU(C) and GPGPU(CUDA) implementations

    Priimak, Dmitri


    We present finite differences numerical algorithm for solving 2D spatially homogeneous Boltzmann transport equation for semiconductor superlattices (SL) subject to time dependant electric field along SL axis and constant perpendicular magnetic field. Algorithm is implemented in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPUs. We compare performance and merits of one implementation versus another and discuss various methods of optimization.

  18. A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions

    Rezaei, M. P.; Zamanian, M.


    In this paper, the influences of nonideal boundary conditions (due to flexibility) on the primary resonant behavior of a piezoelectrically actuated microbeam have been studied, for the first time. The structure has been assumed to treat as an Euler-Bernoulli beam, considering the effects of geometric nonlinearity. In this work, the general nonideal supports have been modeled as a the combination of horizontal, vertical and rotational springs, simultaneously. Allocating particular values to the stiffness of these springs provides the mathematical models for the majority of boundary conditions. This consideration leads to use a two-dimensional analysis of the multiple scales method instead of previous works' method (one-dimensional analysis). If one neglects the nonideal effects, then this paper would be an effort to solve the two-dimensional equations of motion without a need of a combination of these equations using the shortening or stretching effect. Letting the nonideal effects equal to zero and comparing their results with the results of previous approaches have been demonstrated the accuracy of the two-dimensional solutions. The results have been identified the unique effects of constraining and stiffening of boundaries in horizontal, vertical and rotational directions. This means that it is inaccurate to suppose the nonideality of supports only in one or two of these directions like as previous works. The findings are of vital importance as a better prediction of the frequency response for the nonideal supports. Furthermore, the main findings of this effort can help to choose appropriate boundary conditions for desired systems.

  19. Two-Dimensional Toda-Heisenberg Lattice

    Vadim E. Vekslerchik


    Full Text Available We consider a nonlinear model that is a combination of the anisotropic two-dimensional classical Heisenberg and Toda-like lattices. In the framework of the Hirota direct approach, we present the field equations of this model as a bilinear system, which is closely related to the Ablowitz-Ladik hierarchy, and derive its N-soliton solutions.

  20. Mobility anisotropy of two-dimensional semiconductors

    Lang, Haifeng; Zhang, Shuqing; Liu, Zhirong


    The carrier mobility of anisotropic two-dimensional semiconductors under longitudinal acoustic phonon scattering was theoretically studied using deformation potential theory. Based on the Boltzmann equation with the relaxation time approximation, an analytic formula of intrinsic anisotropic mobility was derived, showing that the influence of effective mass on mobility anisotropy is larger than those of deformation potential constant or elastic modulus. Parameters were collected for various anisotropic two-dimensional materials (black phosphorus, Hittorf's phosphorus, BC2N , MXene, TiS3, and GeCH3) to calculate their mobility anisotropy. It was revealed that the anisotropic ratio is overestimated by the previously described method.

  1. 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.

  2. Application of Mixed Differential Quadrature Method for Solving the Coupled Two-Dimensional Incompressible Navier-Stokes Equation and Heat Equation%混合型微分求积法对求解联立的二维不可压Navier-Stokes方程和热方程的应用

    A.S.J.AL-SAIF; 朱正佑


    The traditional differential quadrature method was improved by using the upwind difference scheme for the convectiveterms to solve the coupled two-dimensional incompressible Navier-stokes equations and heat equation. The new method was comparedwith the conventional differential quadrature method in the aspects of convergence and accuracy. The results show that the newmethod is more accurate, and has better convergence than the conventional differential quadrature method for numerically computingthe steady-state solution.

  3. Mean-square exponential stability of the semi-implicit Euler method for stochastic delay differential equations with jumps%带跳随机延迟微分方程半隐式Euler 方法的均方指数稳定性

    徐丽丽; 刘翙


    研究带跳随机延迟微分方程半隐式Euler方法的均方指数稳定性。将半隐式Euler方法应用到维纳过程和泊松过程驱动下的非线性随机延迟微分方程上进行讨论,给出了半隐式Euler方法的均方指数稳定性的条件。%In this paper ,the authors investigated the mean square exponential stability of the semi -implicit Euler method for stochastic delay differential equations with jumps .The semi implicit Euler method applied to the nonlinear stochastic delay dif -ferential equations which driven by Wiener process and Poisson process , and gave conditions about mean square exponential stability of the semi-implicit Euler method .

  4. Euler/Navier-Stokes Methods

    Pulliam, Tom; Kwak, Dochan (Technical Monitor)


    Implicit methods have been the workhorse for the Euler and Navier-Stokes equations for the last 25 years. The ground breaking work of Dr. Joe Steger in implementing such techniques in practical Euler and Navier-Stokes codes provided the basis for all the success in this area. This presentation will highlight his contribution and technical excellence in the area of implicit methods for CFD.

  5. GMS- stability of the composite Euler method for a linear stochastic differential delay equation%线性随机微分延迟方程复合Euler方法的均方稳定性

    周立群; 王薇


    The general mean square (GMS) stability of the composite Euler method for a linear stochastic differential delay equation is investigated. Conditions of the general mean square stability of the composite Euler method for a linear stochastic differential delay equation is given. It is shown that the composite Euler method is GMS - stable under these conditions. The numerical examples are presented to support the theoretical analysis.%研究了复合Euler方法对线性随机微分延迟方程的全局均方稳定性,给出复合Euler方法全局稳定性的条件并证明在这些条件下复合Euler方法是GMS-稳定的,给出数值算例支持理论分析.

  6. 三维等熵欧拉方程组解的爆破%Blow Up of Isentropic Euler Equations in Three Dimensions

    朱旭生; 艾利娜; 汤传扬


    The blow up of the initial value problems of classical solutions of the isentropic Euler equations with damping term in three dimensions was investigated .Under the condition of M(0) <0 , if the initial momentum of cer-tain functions is large enough , then it follows that the classical solution must blow up in a limited time .%研究了带阻尼项的三维等熵欧拉方程组初值问题经典解的爆破。在M(0)<0条件下,若初始动量的某些泛函足够大时,得到了其经典解在有限时间内必定发生爆破的结论。

  7. Mass, momentum and energy conserving (MaMEC) discretizations on general grids for the compressible Euler and shallow water equations

    Hof, Bas van ’t; Veldman, Arthur E.P.


    The paper explains a method by which discretizations of the continuity and momentum equations can be designed, such that they can be combined with an equation of state into a discrete energy equation. The resulting 'MaMEC' discretizations conserve mass, momentum as well as energy, although no explic

  8. Beatification: Flattening the Poisson Bracket for Two-Dimensional Fluid and Plasma Theories

    Viscondi, Thiago F; Morrison, Philip J


    A perturbative method called beatification is presented for a class of two-dimensional fluid and plasma theories. The Hamiltonian systems considered, namely the Euler, Vlasov-Poisson, Hasegawa-Mima, and modified Hasegawa-Mima equations, are naturally described in terms of noncanonical variables. The beatification procedure amounts to finding the correct transformation that removes the explicit variable dependence from a noncanonical Poisson bracket and replaces it with a fixed dependence on a chosen state in phase space. As such, beatification is a major step toward casting the Hamiltonian system in its canonical form, thus enabling or facilitating the use of analytical and numerical techniques that require or favor a representation in terms of canonical, or beatified, Hamiltonian variables.

  9. Topological aspect of disclinations in two-dimensional crystals

    Qi Wei-Kai; Zhu Tao; Chen Yong; Ren Ji-Rong


    By using topological current theory, this paper studies the inner topological structure of disclinations during the melting of two-dimensional systems. From two-dimensional elasticity theory, it finds that there are topological currents for topological defects in homogeneous equation. The evolution of disclinations is studied, and the branch conditions for generating, annihilating, crossing, splitting and merging of disclinations are given.


    partial differential equations . The criteria for computational stability and the truncation errors are studied in connection with application to some simple, linear equations. The method has a built-in damping which increases with decreasing wave-length and increasing wave speed, and seems to be well suited for equations in time and one space coordinate. For equations in time and two space coordinates, however, a complication arises, and a shortening of the time step seems to be necessary in order to secure stability.

  11. Stability of leap-frog constant-coefficients semi-implicit schemes for the fully elastic system of Euler equations. Flat-terrain case

    Benard, P; Vivoda, J; Smolikova, P; Benard, Pierre; Laprise, Rene; Vivoda, Jozef; Smolikova, Petra


    The aim of this paper is to investigate the response of this system/scheme in terms of stability in presence of explicitly treated residual terms, as it inevitably occurs in the reality of NWP. This sudy is restricted to the impact of thermal and baric residual terms (metric residual terms linked to the orography are not considered here). It is shown that conversely to what occurs with Hydrostatic Primitive Equations, the choice of the prognostic variables used to solve the system in time is of primary importance for the robustness with Euler Equations. For an optimal choice of prognostic variables, unconditionnally stable schemes can be obtained (with respect to the length of the time-step), but only for a smaller range of reference states than in the case of Hydrostatic Primitive Equations. This study also indicates that: (i) vertical coordinates based on geometrical height and on mass behave similarly in terms of stability for the problems examined here, and (ii) hybrid coordinates induce an intrinsic inst...

  12. Dynamics of film. [two dimensional continua theory

    Zak, M.


    The general theory of films as two-dimensional continua are elaborated upon. As physical realizations of such a model this paper examines: inextensible films, elastic films, and nets. The suggested dynamic equations have enabled us to find out the characteristic speeds of wave propagation of the invariants of external and internal geometry and formulate the criteria of instability of their shape. Also included herein is a detailed account of the equation describing the film motions beyond the limits of the shape stability accompanied by the formation of wrinkles. The theory is illustrated by examples.

  13. The characters of nonlinear vibration in the two-dimensional discrete monoatomic lattice

    XU Quan; TIAN Qiang


    The two-dimensional discrete monoatomic lattice is analyzed. Taking nearest-neighbor interaction into account, the characters of the nonlinear vibration in two-dimensional discrete monoatomic lattice are described by the two-dimensional cubic nonlinear Schrodinger equation. Considering the quartic nonlinear potential, the two-dimensional discrete-soliton trains and the solutions perturbed by the neck mode are presented.

  14. Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations

    Gassner, Gregor J.; Winters, Andrew R.; Kopriva, David A.


    Fisher and Carpenter (2013) [12] found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux. We show that besides the construction of entropy stable high-order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to prove which of those split forms are at least kinetic energy preserving. With this, we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid Taylor-Green vortex and show that the new split forms enhance the robustness of high-order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.

  15. A new angle on the Euler angles

    Markley, F. Landis; Shuster, Malcolm D.


    We present a generalization of the Euler angles to axes beyond the twelve conventional sets. The generalized Euler axes must satisfy the constraint that the first and the third are orthogonal to the second; but the angle between the first and third is arbitrary, rather than being restricted to the values 0 and pi/2, as in the conventional sets. This is the broadest generalization of the Euler angles that provides a representation of an arbitrary rotation matrix. The kinematics of the generalized Euler angles and their relation to the attitude matrix are presented. As a side benefit, the equations for the generalized Euler angles are universal in that they incorporate the equations for the twelve conventional sets of Euler angles in a natural way.

  16. Hydrodynamic aspects of premixed flame stripes in two-dimensional stagnation-point flows

    Lee, H.; Sohrab, S.H. [Northwestern Univ., Evanston, IL (United States). Dept. of Mechanical Engineering


    The behavior of cellular premixed flames of rich butane-air in the two-dimensional stagnation-point flow configuration has been investigated. It is found that the stretching of the cellular flame results in the alignment f the ridge (extinction) and the trough (combustion) zones of the individual cells such as to form a series of parallel flame stripes. The number of flame stripes as a function of the equivalence ratio for three different mean velocities at the nozzle have been determined. Through the introduction of a generalized form of the stream function periodic velocity fields are obtained as the exact solutions of the Euler equation for the nonreactive finite-jet two-dimensional stagnation flow. The predicted periodic velocity profiles are confirmed by the experimental observation of the streamlines in nonreactive flow made visible by laser-sheet lighting. The observed average size of the flame stripes is found to be in good agreement with the predicted value. Similar periodic velocity profiles are also obtained for the viscous flow within the laminar boundary layer by treatment of the unsteady vorticity equation first described by Taylor. The results support an earlier prediction by Williams that cellular flame structures that are affected mainly by diffusive-thermal phenomena may in fact be initiated by the hydrodynamic instability.

  17. Generalization of the Euler Angles

    Bauer, Frank H. (Technical Monitor); Shuster, Malcolm D.; Markley, F. Landis


    It is shown that the Euler angles can be generalized to axes other than members of an orthonormal triad. As first shown by Davenport, the three generalized Euler axes, hereafter: Davenport axes, must still satisfy the constraint that the first two and the last two axes be mutually perpendicular if these axes are to define a universal set of attitude parameters. Expressions are given which relate the generalized Euler angles, hereafter: Davenport angles, to the 3-1-3 Euler angles of an associated direction-cosine matrix. The computation of the Davenport angles from the attitude matrix and their kinematic equation are presented. The present work offers a more direct development of the Davenport angles than Davenport's original publication and offers additional results.

  18. Bound states of two-dimensional relativistic harmonic oscillators

    Qiang Wen-Chao


    We give the exact normalized bound state wavefunctions and energy expressions of the Klein-Gordon and Dirac equations with equal scalar and vector harmonic oscillator potentials in the two-dimensional space.

  19. Two-dimensional discrete gap breathers in a two-dimensional discrete diatomic Klein-Gordon lattice

    XU Quan; QIANG Tian


    We study the existence and stability of two-dimensional discrete breathers in a two-dimensional discrete diatomic Klein-Gordon lattice consisting of alternating light and heavy atoms, with nearest-neighbor harmonic coupling.Localized solutions to the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum, i.e. two-dimensional gap breathers, are investigated numerically. The numerical results of the corresponding algebraic equations demonstrate the possibility of the existence of two-dimensional gap breathers with three types of symmetries, i.e., symmetric, twin-antisymmetric and single-antisymmetric. Their stability depends on the nonlinear on-site potential (soft or hard), the interaction potential (attractive or repulsive)and the center of the two-dimensional gap breather (on a light or a heavy atom).

  20. Two-dimensional liquid chromatography

    Græsbøll, Rune

    of this thesis is on online comprehensive two-dimensional liquid chromatography (online LC×LC) with reverse phase in both dimensions (online RP×RP). Since online RP×RP has not been attempted before within this research group, a significant part of this thesis consists of knowledge and experience gained...

  1. Euler: programa didáctico de elementos finitos

    Dorian Luis Linero Segrera


    Full Text Available This article explains the characteristics of the Euler software, which was used as a learning tool on finite element method with emphasis on structural analysis. Euler can solve the following problems, among others: static matríx analysis of truss and plane frames; stability analysis, evaluation of frequencies and vibration modes in plane frames, displacement in beams and in elements subjected to axial force and other problems controlled by the one-dimensional field differential equation that is shown in the article. Furthermore, the program can solve: torsion ofnoncircular sections, irrotational flow, heat transfer, and other problems controlled by two-dimensional field differential equation that is shown in the article. The program also allows for solutions to problems of twodimensional elasticity: plane stress or plane strain. In order to operate the program, the user should write, one by one, the necessary instructions to obtain the quantities of interest. The available instructions are classified as follows: matrix edition, basic matrix operations, simultaneous equation systems, matrix and vector assemblage, numeration of freedom degrees, eigenvalues and eigenvectors. Futhermore, the creation of matrix elements as shape functions, gradient vector, stiffness matrix, force vector, interelement contribution, transformation matrix and constant elastic matrix.

  2. Acoustic Wave Propagation Modeling by a Two-dimensional Finite-difference Summation-by-parts Algorithm

    Kim, K. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Petersson, N. A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Rodgers, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)


    Acoustic waveform modeling is a computationally intensive task and full three-dimensional simulations are often impractical for some geophysical applications such as long-range wave propagation and high-frequency sound simulation. In this study, we develop a two-dimensional high-order accurate finite-difference code for acoustic wave modeling. We solve the linearized Euler equations by discretizing them with the sixth order accurate finite difference stencils away from the boundary and the third order summation-by-parts (SBP) closure near the boundary. Non-planar topographic boundary is resolved by formulating the governing equation in curvilinear coordinates following the interface. We verify the implementation of the algorithm by numerical examples and demonstrate the capability of the proposed method for practical acoustic wave propagation problems in the atmosphere.

  3. Weakly disordered two-dimensional Frenkel excitons

    Boukahil, A.; Zettili, Nouredine


    We report the results of studies of the optical properties of weakly disordered two- dimensional Frenkel excitons in the Coherent Potential Approximation (CPA). An approximate complex Green's function for a square lattice with nearest neighbor interactions is used in the self-consistent equation to determine the coherent potential. It is shown that the Density of States is very much affected by the logarithmic singularities in the Green's function. Our CPA results are in excellent agreement with previous investigations by Schreiber and Toyozawa using the Monte Carlo simulation.

  4. Mobility anisotropy of two-dimensional semiconductors

    Lang, Haifeng; Liu, Zhirong


    The carrier mobility of anisotropic two-dimensional (2D) semiconductors under longitudinal acoustic (LA) phonon scattering was theoretically studied with the deformation potential theory. Based on Boltzmann equation with relaxation time approximation, an analytic formula of intrinsic anisotropic mobility was deduced, which shows that the influence of effective mass to the mobility anisotropy is larger than that of deformation potential constant and elastic modulus. Parameters were collected for various anisotropic 2D materials (black phosphorus, Hittorf's phosphorus, BC$_2$N, MXene, TiS$_3$, GeCH$_3$) to calculate their mobility anisotropy. It was revealed that the anisotropic ratio was overestimated in the past.

  5. Degenerate Euler zeta function

    Kim, Taekyun


    Recently, T. Kim considered Euler zeta function which interpolates Euler polynomials at negative integer (see [3]). In this paper, we study degenerate Euler zeta function which is holomorphic function on complex s-plane associated with degenerate Euler polynomials at negative integers.

  6. 二维非线性抛物型方程参数反演的贝叶斯推理估计%The estimated Bayesian inference of two-dimensional nonlinear parabolic equation parameter inversion

    陈亚文; 邹学文


    为了克服观测数据有限以及数据存在一定误差对参数反演结果的影响,提出了一种参数反演的有效算法.根据已知参数的先验分布和已经获得的有误差的监测数据,以贝叶斯推理作为理论基础,获得参数的联合后验概率密度函数,再利用马尔科夫链蒙特卡罗模拟对后验分布进行采样,获得参数的后验边缘概率密度,由此得到了参数的数学期望等有效的统计量.数值模拟结果表明,此算法能够有效地解决二维非线性抛物型方程的参数识别反问题,且具有较高的精度.%In order to overcome the limited observation data with noise, an inversion of the effective parameters algorithm is presented. First, according to the parameters,a priori distribution and the limited observation data with noise, Bayesian inference as a theoretical foundation,parameters of the joint posterior probability density function are obtained. Markov chain Monte Carlo simulation was taken to sample the posterior distribution to get the marginal posterior probability function of the parameters, and the statistical quantities such as the mathematic expectation were calculated. Experimental results show that this algorithm can successfully solve the problem of two-dimensional nonlinear parabolic equation parameter inversion and inversion results have higher accuracy.

  7. Two dimensional unstable scar statistics.

    Warne, Larry Kevin; Jorgenson, Roy Eberhardt; Kotulski, Joseph Daniel; Lee, Kelvin S. H. (ITT Industries/AES Los Angeles, CA)


    This report examines the localization of time harmonic high frequency modal fields in two dimensional cavities along periodic paths between opposing sides of the cavity. The cases where these orbits lead to unstable localized modes are known as scars. This paper examines the enhancements for these unstable orbits when the opposing mirrors are both convex and concave. In the latter case the construction includes the treatment of interior foci.

  8. Two-Dimensional Vernier Scale

    Juday, Richard D.


    Modified vernier scale gives accurate two-dimensional coordinates from maps, drawings, or cathode-ray-tube displays. Movable circular overlay rests on fixed rectangular-grid overlay. Pitch of circles nine-tenths that of grid and, for greatest accuracy, radii of circles large compared with pitch of grid. Scale enables user to interpolate between finest divisions of regularly spaced rule simply by observing which mark on auxiliary vernier rule aligns with mark on primary rule.

  9. Linearized Euler Equations for the Determination of Scattering Matrices for Orifice and Perforated Plate Configurations in the High Mach Number Regime

    Moritz Schulze


    Full Text Available The interaction of a plane acoustic wave and a sheared flow is numerically investigated for simple orifice and perforated plate configurations in an isolated, non-resonant environment for Mach numbers up to choked conditions in the holes. Analytical derivations found in the literature are not valid in this regime due to restrictions to low Mach numbers and incompressible conditions. To allow for a systematic and detailed parameter study, a low-cost hybrid Computational Fluid Dynamic/Computational Aeroacoustic (CFD/CAA methodology is used. For the CFD simulations, a standard k–ϵ Reynolds-Averaged Navier–Stokes (RANS model is employed, while the CAA simulations are based on frequency space transformed linearized Euler equations (LEE, which are discretized in a stabilized Finite Element method. Simulation times in the order of seconds per frequency allow for a detailed parameter study. From the application of the Multi Microphone Method together with the two-source location procedure, acoustic scattering matrices are calculated and compared to experimental findings showing very good agreement. The scattering properties are presented in the form of scattering matrices for a frequency range of 500–1500 Hz.

  10. A hybrid pressure-density-based Mach uniform algorithm for 2D Euler equations on unstructured grids by using multi-moment finite volume method

    Xie, Bin; Deng, Xi; Sun, Ziyao; Xiao, Feng


    We propose a novel Mach-uniform numerical model for 2D Euler equations on unstructured grids by using multi-moment finite volume method. The model integrates two key components newly developed to solve compressible flows on unstructured grids with improved accuracy and robustness. A new variant of AUSM scheme, so-called AUSM+-pcp (AUSM+ with pressure-correction projection), has been devised including a pressure-correction projection to the AUSM+ flux splitting, which maintains the exact numerical conservativeness and works well for all Mach numbers. A novel 3th-order, non-oscillatory and less-dissipative reconstruction has been proposed by introducing a multi-dimensional limiting and a BVD (boundary variation diminishing) treatment to the VPM (volume integrated average (VIA) and point value (PV) based multi-moment) reconstruction. The resulting reconstruction scheme, the limited VPM-BVD formulation, is able to resolve both smooth and non-smooth solutions with high fidelity. Benchmark tests have been used to verify the present model. The numerical results substantiate the present model as an accurate and robust unstructured-grid formulation for flows of all Mach numbers.

  11. 关于F.Smarandache函数和欧拉函数的三个方程%Three equations involving F.Smarandache function and Euler function



    For any positive integer n, SL(n) is Smarandache LCM function,φ(n) is Euler function, S(n) is Smarandache function. Yanlin Zhao studied all positive integer solutions of φ(n) = S(nk) and SL(n) =φ(n). The main purpose is to study the solvability of three equations φ(nk) = S(n) (k≥2) , SL(nk) =φ(n) (k≥2) and SL(n) =φ(nk) (k≥2) by using the property of SL(n) ,φ(n) ,S(n) and the elementary method. All positive integer solutions of them are given.%对任意正整数n,S(n)为Smarandache LCM函数,φ(n)为欧拉函数,S(n)为Smarandache函数,赵艳琳研究了方程φ(n)=S(nk)(k为任意正整数)与方程SL(n)=φ(n)的所有正整数解.利用SL(n),φ(n),S(n)的性质结合初等方法研究了三类方程φ(nk)=S(n)(l≥2),SL(nk)=φ(n)(k≥2)与SL(n)=φ(nk)(k≥2)的可解性问题并求出所有正整数解.

  12. New developments in the method of space-time conservation element and solution element: Applications to the Euler and Navier-Stokes equations

    Chang, Sin-Chung


    A new numerical framework for solving conservation laws is being developed. This new approach differs substantially in both concept and methodology from the well-established methods--i.e., finite difference, finite volume, finite element, and spectral methods. It is conceptually simple and designed to avoid several key limitations to the above traditional methods. An explicit model scheme for solving a simple 1-D unsteady convection-diffusion equation is constructed and used to illuminate major differences between the current method and those mentioned above. Unexpectedly, its amplification factors for the pure convection and pure diffusion cases are identical to those of the Leapfrog and the DuFort-Frankel schemes, respectively. Also, this explicit scheme and its Navier-Stokes extension have the unusual property that their stabilities are limited only by the CFL condition. Moreover, despite the fact that it does not use any flux-limiter or slope-limiter, the Navier-Stokes solver is capable of generating highly accurate shock tube solutions with shock discontinuities being resolved within one mesh interval. An accurate Euler solver also is constructed through another extension. It has many unusual properties, e.g., numerical diffusion at all mesh points can be controlled by a set of local parameters.

  13. Euler's Three-Body Problem.

    Wild, Walter J.


    Discusses the simplest three-body problem, known as Euler's problem. The article, intended for students in the undergraduate mathematics and physics curricula, shows how the complex equations for a specific three-body problem can be solved on a small calculator. (HM)

  14. Two-dimensional liquid chromatography

    Græsbøll, Rune

    Two-dimensional liquid chromatography has received increasing interest due to the rise in demand for analysis of complex chemical mixtures. Separation of complex mixtures is hard to achieve as a simple consequence of the sheer number of analytes, as these samples might contain hundreds or even...... dimensions. As a consequence of the conclusions made within this thesis, the research group has, for the time being, decided against further development of online LC×LC systems, since it was not deemed ideal for the intended application, the analysis of the polar fraction of oil. Trap-and...

  15. The Method of Space-Time Conservation Element and Solution Element—A New Approach for Solving the Navier-Stokes and Euler Equations

    Chang, Sin-Chung


    generally cannot be extended to solve the Euler equations. Thus, the inviscid version is modified. Stability of this modified scheme, referred to as the a-ε scheme, is limited by the CFL condition and 0 ≤ ε ≤ 1, where ε is a special parameter that controls numerical dissipation. Moreover, if ε = 0, the amplification factors of the a-ε scheme are identical to those of the Leapfrog scheme, which has no numerical dissipation. On the other hand, if ε = 1, the two amplification factors of the a-ε scheme become the same function of the Courant number and the phase angle. Unexpectedly, this function also is the amplification factor of the highly diffusive Lax scheme. Note that, because the Lax scheme is very diffusive and it uses a mesh that is staggered in time, a two-level scheme using such a mesh is often associated with a highly diffusive scheme. The a-ε scheme, which also uses a mesh staggering in time, demonstrates that it can also be a scheme with no numerical dissipation. The Euler extension of the a -ε scheme has stability conditions similar to those of the a -epsiv; scheme itself. It has the unusual property that numerical dissipation at all mesh points can be controlled by a set of local parameters, Moreover, it is capable of generating accurate shock tube solutions with the CFL number ranging from close to 1 to 0.022

  16. Nonlinear excitations in two-dimensional molecular structures with impurities

    Gaididei, Yuri Borisovich; Rasmussen, Kim; Christiansen, Peter Leth


    We study the nonlinear dynamics of electronic excitations interacting with acoustic phonons in two-dimensional molecular structures with impurities. We show that the problem is reduced to the nonlinear Schrodinger equation with a varying coefficient. The latter represents the influence of the imp......We study the nonlinear dynamics of electronic excitations interacting with acoustic phonons in two-dimensional molecular structures with impurities. We show that the problem is reduced to the nonlinear Schrodinger equation with a varying coefficient. The latter represents the influence...... excitations. Analytical results are in good agreement with numerical simulations of the nonlinear Schrodinger equation....

  17. On the continua in two-dimensional nonadiabatic magnetohydrodynamic spectra

    De Ploey, A.; Van der Linden, R. A. M.; Belien, A. J. C.


    The equations for the continuous subspectra of the linear magnetohydrodynamic (MHD) normal modes spectrum of two-dimensional (2D) plasmas are derived in general curvilinear coordinates, taking nonadiabatic effects in the energy equation into account. Previously published derivations of continuous sp

  18. Generalizations of Euler Numbers and Euler Numbers of Higher Order

    LUOQiu-ming; QIFeng


    The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.

  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. Euler/Navier-Stokes Solvers Applied to Ducted Fan Configurations

    Keith, Theo G., Jr.; Srivastava, Rakesh


    Due to noise considerations, ultra high bypass ducted fans have become a more viable design. These ducted fans typically consist of a rotor stage containing a wide chord fan and a stator stage. One of the concerns for this design is the classical flutter that keeps occurring in various unducted fan blade designs. These flutter are catastrophic and are to be avoided in the flight envelope of the engine. Some numerical investigations by Williams, Cho and Dalton, have suggested that a duct around a propeller makes it more unstable. This needs to be further investigated. In order to design an engine to safely perform a set of desired tasks, accurate information of the stresses on the blade during the entire cycle of blade motion is required. This requirement in turn demands that accurate knowledge of steady and unsteady blade loading be available. Aerodynamic solvers based on unsteady three-dimensional analysis will provide accurate and fast solutions and are best suited for aeroelastic analysis. The Euler solvers capture significant physics of the flowfield and are reasonably fast. An aerodynamic solver Ref. based on Euler equations had been developed under a separate grant from NASA Lewis in the past. Under the current grant, this solver has been modified to calculate the aeroelastic characteristics of unducted and ducted rotors. Even though, the aeroelastic solver based on three-dimensional Euler equations is computationally efficient, it is still very expensive to investigate the effects of multiple stages on the aeroelastic characteristics. In order to investigate the effects of multiple stages, a two-dimensional multi stage aeroelastic solver was also developed under this task, in collaboration with Dr. T. S. R. Reddy of the University of Toledo. Both of these solvers were applied to several test cases and validated against experimental data, where available.

  1. Two-dimensional quantum repeaters

    Wallnöfer, J.; Zwerger, M.; Muschik, C.; Sangouard, N.; Dür, W.


    The endeavor to develop quantum networks gave rise to a rapidly developing field with far-reaching applications such as secure communication and the realization of distributed computing tasks. This ultimately calls for the creation of flexible multiuser structures that allow for quantum communication between arbitrary pairs of parties in the network and facilitate also multiuser applications. To address this challenge, we propose a two-dimensional quantum repeater architecture to establish long-distance entanglement shared between multiple communication partners in the presence of channel noise and imperfect local control operations. The scheme is based on the creation of self-similar multiqubit entanglement structures at growing scale, where variants of entanglement swapping and multiparty entanglement purification are combined to create high-fidelity entangled states. We show how such networks can be implemented using trapped ions in cavities.

  2. Two-dimensional capillary origami

    Brubaker, N. D.; Lega, J.


    We describe a global approach to the problem of capillary origami that captures all unfolded equilibrium configurations in the two-dimensional setting where the drop is not required to fully wet the flexible plate. We provide bifurcation diagrams showing the level of encapsulation of each equilibrium configuration as a function of the volume of liquid that it contains, as well as plots representing the energy of each equilibrium branch. These diagrams indicate at what volume level the liquid drop ceases to be attached to the endpoints of the plate, which depends on the value of the contact angle. As in the case of pinned contact points, three different parameter regimes are identified, one of which predicts instantaneous encapsulation for small initial volumes of liquid.

  3. Two-dimensional cubic convolution.

    Reichenbach, Stephen E; Geng, Frank


    The paper develops two-dimensional (2D), nonseparable, piecewise cubic convolution (PCC) for image interpolation. Traditionally, PCC has been implemented based on a one-dimensional (1D) derivation with a separable generalization to two dimensions. However, typical scenes and imaging systems are not separable, so the traditional approach is suboptimal. We develop a closed-form derivation for a two-parameter, 2D PCC kernel with support [-2,2] x [-2,2] that is constrained for continuity, smoothness, symmetry, and flat-field response. Our analyses, using several image models, including Markov random fields, demonstrate that the 2D PCC yields small improvements in interpolation fidelity over the traditional, separable approach. The constraints on the derivation can be relaxed to provide greater flexibility and performance.

  4. Euler, the Master Calculator.

    Taylor, Jerry D.


    Provides a brief, condensed biography of the eighteenth-century mathematician, Leonard Euler, focusing on some of his contributions to mathematics. Also presents several problems and suggests how Euler might have solved them. (JN)

  5. Two-dimensional gauge theoretic supergravities

    Cangemi, D.; Leblanc, M.


    We investigate two-dimensional supergravity theories, which can be built from a topological and gauge invariant action defined on an ordinary surface. One is the N = 1 supersymmetric extension of the Jackiw-Teitelboim model presented by Chamseddine in a superspace formalism. We complement the proof of Montano, Aoaki and Sonnenschein that this extension is topological and gauge invariant, based on the graded de Sitter algebra. Not only do the equations of motion correspond to the supergravity ones and do gauge transformations encompass local supersymmetries, but we also identify the ∫-theory with the superfield formalism action written by Chamseddine. Next, we show that the N = 1 supersymmetric extension of string-inspired two-dimensional dilaton gravity put forward by Park and Strominger cannot be written as a ∫-theory. As an alternative, we propose two topological and gauge theories that are based on a graded extension of the extended Poincaré algebra and satisfy a vanishing-curvature condition. Both models are supersymmetric extensions of the string-inspired dilaton gravity.




    A universal variational formulation for two dimensional fluid mechanics is obtained, which is subject to the so-called parameter-constrained equations (the relationship between parameters in two governing equations). By eliminating the constraints, the generalized variational principle (GVPs) can be readily derived from the formulation. The formulation can be applied to any conditions in case the governing equations can be converted into conservative forms. Some illustrative examples are given to testify the effectiveness and simplicity of the method.

  7. Quasinormal frequencies of asymptotically flat two-dimensional black holes

    Lopez-Ortega, A


    We discuss whether the minimally coupled massless Klein-Gordon and Dirac fields have well defined quasinormal modes in single horizon, asymptotically flat two-dimensional black holes. To get the result we solve the equations of motion in the massless limit and we also calculate the effective potentials of Schrodinger type equations. Furthermore we calculate exactly the quasinormal frequencies of the Dirac field propagating in the two-dimensional uncharged Witten black hole. We compare our results on its quasinormal frequencies with other already published.

  8. Phonon hydrodynamics in two-dimensional materials.

    Cepellotti, Andrea; Fugallo, Giorgia; Paulatto, Lorenzo; Lazzeri, Michele; Mauri, Francesco; Marzari, Nicola


    The conduction of heat in two dimensions displays a wealth of fascinating phenomena of key relevance to the scientific understanding and technological applications of graphene and related materials. Here, we use density-functional perturbation theory and an exact, variational solution of the Boltzmann transport equation to study fully from first-principles phonon transport and heat conductivity in graphene, boron nitride, molybdenum disulphide and the functionalized derivatives graphane and fluorographene. In all these materials, and at variance with typical three-dimensional solids, normal processes keep dominating over Umklapp scattering well-above cryogenic conditions, extending to room temperature and more. As a result, novel regimes emerge, with Poiseuille and Ziman hydrodynamics, hitherto typically confined to ultra-low temperatures, characterizing transport at ordinary conditions. Most remarkably, several of these two-dimensional materials admit wave-like heat diffusion, with second sound present at room temperature and above in graphene, boron nitride and graphane.

  9. A parametrization of two-dimensional turbulence based on a maximum entropy production principle with a local conservation of energy

    Chavanis, Pierre-Henri, E-mail: [Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse (France)


    In the context of two-dimensional (2D) turbulence, we apply the maximum entropy production principle (MEPP) by enforcing a local conservation of energy. This leads to an equation for the vorticity distribution that conserves all the Casimirs, the energy, and that increases monotonically the mixing entropy (H-theorem). Furthermore, the equation for the coarse-grained vorticity dissipates monotonically all the generalized enstrophies. These equations may provide a parametrization of 2D turbulence. They do not generally relax towards the maximum entropy state. The vorticity current vanishes for any steady state of the 2D Euler equation. Interestingly, the equation for the coarse-grained vorticity obtained from the MEPP turns out to coincide, after some algebraic manipulations, with the one obtained with the anticipated vorticity method. This shows a connection between these two approaches when the conservation of energy is treated locally. Furthermore, the newly derived equation, which incorporates a diffusion term and a drift term, has a nice physical interpretation in terms of a selective decay principle. This sheds new light on both the MEPP and the anticipated vorticity method. (paper)

  10. Classifying Two-dimensional Hyporeductive Triple Algebras

    Issa, A Nourou


    Two-dimensional real hyporeductive triple algebras (h.t.a.) are investigated. A classification of such algebras is presented. As a consequence, a classification of two-dimensional real Lie triple algebras (i.e. generalized Lie triple systems) and two-dimensional real Bol algebras is given.

  11. Dynamic effects on the transition between two-dimensional regular and Mach reflection of shock waves in an ideal, steady supersonic free stream

    Naidoo, K


    Full Text Available et al. (1999) investigated the effect of continuous rapid wedge rotation on the point of transition with Euler CFD on moving meshes. In contrast to the work by Markelov et al. (1999), Khotyanovsky et al. (1999) considered larger move- ments... between the three-dimensional Euler CFD predictions of Ivanov et al. (2001) and their measurements from experiments with the finite aspect ratio wedge. This agreement established confidence in their two-dimensional Mach stem predictions with Euler CFD...

  12. Euler as Physicist

    Suisky, Dieter


    "Euler as Physicist" analyzes the exceptional role of Leonhard Euler (1707 - 1783) in the history of science and emphasizes especially his fundamental contributions to physics. Although Euler is famous as the leading mathematician of the 18th century, his contributions to physics are as important for their innovative methods and solutions. Several books are devoted to Euler as mathematician, but none to Euler as physicist, like in this book. Euler’s contributions to mechanics are rooted in his life-long plan presented in two volume treatise programmatically entitled "Mechanics or the science of motion analytically demonstrated". Published in 1736, Euler’s treatise indicates the turn over from the traditional geometric representation of mechanics to a new approach. In writing Mechanics Euler did the first step to put the plan and his completion into practice through 1760. It is of particular interest to study how Euler made immediate use of his mathematics for mechanics and coordinated his progress in math...

  13. Two-dimensional function photonic crystals

    Wu, Xiang-Yao; Liu, Xiao-Jing; Liang, Yu


    In this paper, we have firstly proposed two-dimensional function photonic crystals, which the dielectric constants of medium columns are the functions of space coordinates $\\vec{r}$, it is different from the two-dimensional conventional photonic crystals constituting by the medium columns of dielectric constants are constants. We find the band gaps of two-dimensional function photonic crystals are different from the two-dimensional conventional photonic crystals, and when the functions form of dielectric constants are different, the band gaps structure should be changed, which can be designed into the appropriate band gaps structures by the two-dimensional function photonic crystals.

  14. 翼型采用近似边界条件的欧拉方程数值解%Solution of the Euler Equations with Approximate Boundary Conditions for Thin Airfoils


    This paper presents an efficient numerical method for solving the Euler equations on rectilinear grids. Wall boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the airfoil chord line, which is placed along a grid line. However, the method is not restricted to flows with small disturbances since there are no restrictions on the magnitude of the velocity or pressure perturbations. The mathematical formulation and the numerical implementation of the wall boundary conditions in a finite-volume Euler code are described. Steady transonic flows are calculated about the NACA 0006, NACA 0012 and NACA 0015 airfoils, corresponding to thickness ratios of 6%, 12%, and 15%, respectively. The computed results, including surface pressure distributions, the lift coefficient, the wave drag coefficient, and the pitching moment coefficient, at angles of attack from 0° to 8° are compared with solutions at the same conditions by FLO52, a well-established Euler code using body-fitted curvilinear grids. Results demonstrate that the method yields acceptable accuracies even for the relatively thick NACA 0015 airfoil and at high angles of attack. This study establishes the potential of extending the method to computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in both computer time and human work since it would not require the generation of time-dependent body-fitted grids.

  15. Two-dimensional nonlinear nonequilibrium kinetic theory under steady heat conduction.

    Hyeon-Deuk, Kim


    The two-dimensional steady-state Boltzmann equation for hard-disk molecules in the presence of a temperature gradient has been solved explicitly to second order in density and the temperature gradient. The two-dimensional equation of state and some physical quantities are calculated from it and compared with those for the two-dimensional steady-state Bhatnagar-Gross-Krook equation and information theory. We have found that the same kind of qualitative differences as the three-dimensional case among these theories still appear in the two-dimensional case.

  16. Two-dimensional effects in nonlinear Kronig-Penney models

    Gaididei, Yuri Borisovich; Christiansen, Peter Leth; Rasmussen, Kim


    An analysis of two-dimensional (2D) effects in the nonlinear Kronig-Penney model is presented. We establish an effective one-dimensional description of the 2D effects, resulting in a set of pseudodifferential equations. The stationary states of the 2D system and their stability is studied...

  17. Statistical study of approximations to two dimensional inviscid turbulence

    Glaz, H.M.


    A numerical technique is developed for studying the ergodic and mixing hypotheses for the dynamical systems arising from the truncated Fourier transformed two-dimensional inviscid Navier-Stokes equations. This method has the advantage of exactly conserving energy and entropy (i.e., total vorticity) in the inviscid case except for numerical error in solving the ordinary differential equations. The development of the mathematical model as an approximation to a real physical (turbulent) flow and the numerical results obtained are discussed.

  18. Epi-two-dimensional flow and generalized enstrophy

    Yoshida, Zensho


    The conservation of the enstrophy ($L^2$ norm of the vorticity $\\omega$) plays an essential role in the physics and mathematics of two-dimensional (2D) Euler fluids. Generalizing to compressible ideal (inviscid and barotropic) fluids, the generalized enstrophy $\\int_{\\Sigma(t)} f(\\omega/\\rho)\\rho\\, d^2 x$, ($f$ an arbitrary smooth function, $\\rho$ the density, and $\\Sigma(t)$ an arbitrary 2D domain co-moving with the fluid) is a constant of motion, and plays the same role. On the other hand, for the three-dimensional (3D) ideal fluid, the helicity $\\int_{M} {V}\\cdot\\omega\\,d^3x$, ($V$ the flow velocity, $\\omega=\

  19. Topological defects in two-dimensional crystals

    Chen, Yong; Qi, Wei-Kai


    By using topological current theory, we study the inner topological structure of the topological defects in two-dimensional (2D) crystal. We find that there are two elementary point defects topological current in two-dimensional crystal, one for dislocations and the other for disclinations. The topological quantization and evolution of topological defects in two-dimensional crystals are discussed. Finally, We compare our theory with Brownian-dynamics simulations in 2D Yukawa systems.

  20. Singular analysis of two-dimensional bifurcation system


    Bifurcation properties of two-dimensional bifurcation system are studied in this paper.Universal unfolding and transition sets of the bifurcation equations are obtained.The whole parametric plane is divided into several different persistent regions according to the type of motion,and the different qualitative bifurcation diagrams in different persistent regions are given.The bifurcation properties of the two-dimensional bifurcation system are compared with its reduced one-dimensional system.It is found that the system which is reduced to one dimension has lost many bifurcation properties.

  1. Dynamics of vortex interactions in two-dimensional flows

    Juul Rasmussen, J.; Nielsen, A.H.; Naulin, V.


    a critical value, a(c). Using the Weiss-field, a(c) is estimated for vortex patches. Introducing an effective radius for vortices with distributed vorticity, we find that 3.3 a(c) ...The dynamics and interaction of like-signed vortex structures in two dimensional flows are investigated by means of direct numerical solutions of the two-dimensional Navier-Stokes equations. Two vortices with distributed vorticity merge when their distance relative to their radius, d/R-0l. is below...

  2. On two-dimensional magnetic reconnection with nonuniform resistivity

    Malyshkin, Leonid M.; Kulsrud, Russell M.


    In this paper, two theoretical approaches for the calculation of the rate of quasi-stationary, two-dimensional magnetic reconnection with nonuniform anomalous resistivity are considered in the framework of incompressible magnetohydrodynamics (MHD). In the first, 'global' equations approach, the MHD equations are approximately solved for a whole reconnection layer, including the upstream and downstream regions and the layer center. In the second, 'local' equations approach, the equations are solved across the reconnection layer, including only the upstream region and the layer center. Both approaches give the same approximate answer for the reconnection rate. Our theoretical model is in agreement with the results of recent simulations of reconnection with spatially nonuniform resistivity.

  3. Ground-state and dynamical properties of two-dimensional dipolar Fermi liquids

    Abedinpour, Saeed H.; Asgari, Reza; Tanatar, B.; Polini, Marco


    We study the ground-state properties of a two-dimensional spin-polarized fluid of dipolar fermions within the Euler-Lagrange Fermi-hypernetted-chain approximation. Our method is based on the solution of a scattering Schrödinger equation for the "pair amplitude" g(r), where g(r) is the pair distribution function. A key ingredient in our theory is the effective pair potential, which includes a bosonic term from Jastrow-Feenberg correlations and a fermionic contribution from kinetic energy and exchange, which is tailored to reproduce the Hartree-Fock limit at weak coupling. Very good agreement with recent results based on quantum Monte Carlo simulations is achieved over a wide range of coupling constants up to the liquid-to-crystal quantum phase transition. Using the fluctuation-dissipation theorem and a static approximation for the effective inter-particle interactions, we calculate the dynamical density-density response function, and furthermore demonstrate that an undamped zero-sound mode exists for any value of the interaction strength, down to infinitesimally weak couplings.

  4. Strongly interacting two-dimensional Dirac fermions

    Lim, L.K.; Lazarides, A.; Hemmerich, Andreas; de Morais Smith, C.


    We show how strongly interacting two-dimensional Dirac fermions can be realized with ultracold atoms in a two-dimensional optical square lattice with an experimentally realistic, inherent gauge field, which breaks time reversal and inversion symmetries. We find remarkable phenomena in a temperature

  5. Topology optimization of two-dimensional waveguides

    Jensen, Jakob Søndergaard; Sigmund, Ole


    In this work we use the method of topology optimization to design two-dimensional waveguides with low transmission loss.......In this work we use the method of topology optimization to design two-dimensional waveguides with low transmission loss....

  6. Mapping two-dimensional polar active fluids to two-dimensional soap and one-dimensional sandblasting

    Chen, Leiming; Lee, Chiu Fan; Toner, John


    Active fluids and growing interfaces are two well-studied but very different non-equilibrium systems. Each exhibits non-equilibrium behaviour distinct from that of their equilibrium counterparts. Here we demonstrate a surprising connection between these two: the ordered phase of incompressible polar active fluids in two spatial dimensions without momentum conservation, and growing one-dimensional interfaces (that is, the 1+1-dimensional Kardar-Parisi-Zhang equation), in fact belong to the same universality class. This universality class also includes two equilibrium systems: two-dimensional smectic liquid crystals, and a peculiar kind of constrained two-dimensional ferromagnet. We use these connections to show that two-dimensional incompressible flocks are robust against fluctuations, and exhibit universal long-ranged, anisotropic spatio-temporal correlations of those fluctuations. We also thereby determine the exact values of the anisotropy exponent ζ and the roughness exponents χx,y that characterize these correlations.

  7. Smoothed Particle Hydrodynamics Method for Two-dimensional Stefan Problem

    Tarwidi, Dede


    Smoothed particle hydrodynamics (SPH) is developed for modelling of melting and solidification. Enthalpy method is used to solve heat conduction equations which involved moving interface between phases. At first, we study the melting of floating ice in the water for two-dimensional system. The ice objects are assumed as solid particles floating in fluid particles. The fluid and solid motion are governed by Navier-Stokes equation and basic rigid dynamics equation, respectively. We also propose a strategy to separate solid particles due to melting and solidification. Numerical results are obtained and plotted for several initial conditions.

  8. A new Green's function Monte Carlo algorithm for the solution of the two-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of the communication breakdown problem in space vehicles during re-entry

    Chatterjee, Kausik, E-mail: [Strategic and Military Space Division, Space Dynamics Laboratory, North Logan, UT 84341 (United States); Center for Atmospheric and Space Sciences, Utah State University, Logan, UT 84322 (United States); Roadcap, John R., E-mail: [Air Force Research Laboratory, Kirtland AFB, NM 87117 (United States); Singh, Surendra, E-mail: [Department of Electrical Engineering, The University of Tulsa, Tulsa, OK 74104 (United States)


    The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

  9. A numerical model for density-and-viscosity-dependent flows in two-dimensional variably saturated porous media

    Boufadel, Michel C.; Suidan, Makram T.; Venosa, Albert D.


    We present a formulation for water flow and solute transport in two-dimensional variably saturated media that accounts for the effects of the solute on water density and viscosity. The governing equations are cast in a dimensionless form that depends on six dimensionless groups of parameters. These equations are discretized in space using the Galerkin finite element formulation and integrated in time using the backward Euler scheme with mass lumping. The modified Picard method is used to linearize the water flow equation. The resulting numerical model, the MARUN model, is verified by comparison to published numerical results. It is then used to investigate beach hydraulics at seawater concentration (about 30 g l -1) in the context of nutrients delivery for bioremediation of oil spills on beaches. Numerical simulations that we conducted in a rectangular section of a hypothetical beach revealed that buoyancy in the unsaturated zone is significant in soils that are fine textured, with low anisotropy ratio, and/or exhibiting low physical dispersion. In such situations, application of dissolved nutrients to a contaminated beach in a freshwater solution is superior to their application in a seawater solution. Concentration-engendered viscosity effects were negligible with respect to concentration-engendered density effects for the cases that we considered.

  10. Inductively generating Euler diagrams.

    Stapleton, Gem; Rodgers, Peter; Howse, John; Zhang, Leishi


    Euler diagrams have a wide variety of uses, from information visualization to logical reasoning. In all of their application areas, the ability to automatically layout Euler diagrams brings considerable benefits. In this paper, we present a novel approach to Euler diagram generation. We develop certain graphs associated with Euler diagrams in order to allow curves to be added by finding cycles in these graphs. This permits us to build Euler diagrams inductively, adding one curve at a time. Our technique is adaptable, allowing the easy specification, and enforcement, of sets of well-formedness conditions; we present a series of results that identify properties of cycles that correspond to the well-formedness conditions. This improves upon other contributions toward the automated generation of Euler diagrams which implicitly assume some fixed set of well-formedness conditions must hold. In addition, unlike most of these other generation methods, our technique allows any abstract description to be drawn as an Euler diagram. To establish the utility of the approach, a prototype implementation has been developed.

  11. Two-dimensional imaginary lobachevsky space. Separation of variables and contractions

    Pogosyan, G. S., E-mail:; Yakhno, A. [Universidad de Guadalajara, Departamento de Matematicas, CUCEI (Mexico)


    The Inoenue-Wigner contraction from the SO(2, 1) group to the E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the corresponding two-dimensional homogeneous spaces: two-dimensional one sheeted hyperboloid and two-dimensional pseudo-Euclidean space. Here we consider the contraction limits of some basis functions for the subgroup coordinates only.

  12. Theory and applications of the problem of Euler elastica

    Zelikin, Mikhail I [Steklov Mathematical Institute, Russian Academy of Sciences, Moscow (Russian Federation)


    The paper is devoted to the theory of extremal problems on Euler elastica. The Riccati equation method is used to study sufficient optimality conditions for the associated problem of minimization of the energy of a physical pendulum. Numerous applications are described for the problem of Euler elastica, and its connections with the theory of completely integrable Hamiltonian systems are discussed. Bibliography: 10 titles.

  13. Euler-Poincare Reduction of Externall Forced Rigid Body Motion

    Wisniewski, Rafal; Kulczycki, P.


    . Assuming that the system possesses symmetry and the configuration manifold corresponds to a Lie group, the Euler-Poincaré reduction breaks up the motion into separate equations of dynamics and kinematics. This becomes of particular interest for modelling, estimation and control of mechanical systems......-known Euler-Poincaré reduction to a rigid body motion with forcing....

  14. Euler-Poincaré Reduction of a Rigid Body Motion

    Wisniewski, Rafal; Kulczycki, P.


    . Assuming that the system possesses symmetry and the configuration manifold corresponds to a Lie group, the Euler-Poincaré reduction breaks up the motion into separate equations of dynamics and kinematics. This becomes of particular interest for modelling, estimation and control of mechanical systems......-known Euler-Poincaré reduction to a rigid body motion with forcing....

  15. Euler-Poincare Reduction of a Rigid Body Motion

    Wisniewski, Rafal; Kulczycki, P.


    . Assuming that the system possesses symmetry and the configuration manifold corresponds to a Lie group, the Euler-Poincare reduction breaks up the motion into separate equations of dynamics and kinematics. This becomes of particular interest for modeling, estimation and control of mechanical systems......-known Euler-Poincare reduction to a rigid body motion with forcing....

  16. Two Dimensional Plasmonic Cavities on Moire Surfaces

    Balci, Sinan; Kocabas, Askin; Karabiyik, Mustafa; Kocabas, Coskun; Aydinli, Atilla


    We investigate surface plasmon polariton (SPP) cavitiy modes on two dimensional Moire surfaces in the visible spectrum. Two dimensional hexagonal Moire surface can be recorded on a photoresist layer using Interference lithography (IL). Two sequential exposures at slightly different angles in IL generate one dimensional Moire surfaces. Further sequential exposure for the same sample at slightly different angles after turning the sample 60 degrees around its own axis generates two dimensional hexagonal Moire cavity. Spectroscopic reflection measurements have shown plasmonic band gaps and cavity states at all the azimuthal angles (omnidirectional cavity and band gap formation) investigated. The plasmonic band gap edge and the cavity states energies show six fold symmetry on the two dimensional Moire surface as measured in reflection measurements.

  17. Two-dimensional function photonic crystals

    Liu, Xiao-Jing; Liang, Yu; Ma, Ji; Zhang, Si-Qi; Li, Hong; Wu, Xiang-Yao; Wu, Yi-Heng


    In this paper, we have studied two-dimensional function photonic crystals, in which the dielectric constants of medium columns are the functions of space coordinates , that can become true easily by electro-optical effect and optical kerr effect. We calculated the band gap structures of TE and TM waves, and found the TE (TM) wave band gaps of function photonic crystals are wider (narrower) than the conventional photonic crystals. For the two-dimensional function photonic crystals, when the dielectric constant functions change, the band gaps numbers, width and position should be changed, and the band gap structures of two-dimensional function photonic crystals can be adjusted flexibly, the needed band gap structures can be designed by the two-dimensional function photonic crystals, and it can be of help to design optical devices.

  18. Two-Dimensional Planetary Surface Lander

    Hemmati, H.; Sengupta, A.; Castillo, J.; McElrath, T.; Roberts, T.; Willis, P.


    A systems engineering study was conducted to leverage a new two-dimensional (2D) lander concept with a low per unit cost to enable scientific study at multiple locations with a single entry system as the delivery vehicle.

  19. Conductivity of a two-dimensional guiding center plasma.

    Montgomery, D.; Tappert, F.


    The Kubo method is used to calculate the electrical conductivity of a two-dimensional, strongly magnetized plasma. The particles interact through (logarithmic) electrostatic potentials and move with their guiding center drift velocities (Taylor-McNamara model). The thermal equilibrium dc conductivity can be evaluated analytically, but the ac conductivity involves numerical solution of a differential equation. Both conductivities fall off as the inverse first power of the magnetic field strength.

  20. Interior design of a two-dimensional semiclassic black hole

    Levanony, Dana; 10.1103/PhysRevD.80.084008


    We look into the inner structure of a two-dimensional dilatonic evaporating black hole. We establish and employ the homogenous approximation for the black-hole interior. The field equations admit two types of singularities, and their local asymptotic structure is investigated. One of these singularities is found to develop, as a spacelike singularity, inside the black hole. We then study the internal structure of the evaporating black hole from the horizon to the singularity.

  1. Towards a two dimensional model of surface piezoelectricity

    Monge Víllora, Oscar


    We want to understand the behaviour of flexoelectricity and surface piezoelectricity and distinguish them in order to go deep into the controversies of the filed. This motivate the construction of a model of continuum flexoelectric theory. The model proposed is a two-dimensional model that integrates the electromechanical equations that include the elastic, dielectric, piezoelectric and flexoelectric effect on a rectangular sample. As the flexoelectric and the surface piezoelectric effects ap...

  2. Illustrating the Euler Line.

    Rubillo, James M.


    Euler's discovery about the centroid of a triangle trisecting the line segment joining its circumference to its orthocenter is discussed. An activity that will help students review fundamental concepts is included. (MNS)

  3. Nonclassical Symmetry Analysis of Heated Two-Dimensional Flow Problems

    Naeem, Imran; Naz, Rehana; Khan, Muhammad Danish


    This article analyses the nonclassical symmetries and group invariant solution of boundary layer equations for two-dimensional heated flows. First, we derive the nonclassical symmetry determining equations with the aid of the computer package SADE. We solve these equations directly to obtain nonclassical symmetries. We follow standard procedure of computing nonclassical symmetries and consider two different scenarios, ξ1≠0 and ξ1=0, ξ2≠0. Several nonclassical symmetries are reported for both scenarios. Furthermore, numerous group invariant solutions for nonclassical symmetries are derived. The similarity variables associated with each nonclassical symmetry are computed. The similarity variables reduce the system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) in terms of similarity variables. The reduced system of ODEs are solved to obtain group invariant solution for governing boundary layer equations for two-dimensional heated flow problems. We successfully formulate a physical problem of heat transfer analysis for fluid flow over a linearly stretching porous plat and, with suitable boundary conditions, we solve this problem.

  4. Integral transformation of the Navier-Stokes equations for laminar flow in channels of arbitrary two-dimensional geometry; Transformacao integral das equacoes de Navier-Stokes para escoamento laminar em canais de geometria bidimensional arbitraria

    Perez Guerrero, Jesus Salvador


    Laminar developing flow in channels of arbitrary geometry was studied by solving the Navier-Stokes equations in the stream function-only formulation through the Generalized Integral Transform Technique (GITT). The stream function is expanded in an infinite system based on eigenfunctions obtained by considering solely the diffusive terms of the original formulation. The Navier-Stokes equations are transformed into an infinite system of ordinary differential equations, by using the transformation and inversion formulae. For computational purposes, the infinite series is truncated, according to an automatic error control procedure. The ordinary differential is solved through well-established scientific subroutines from widely available mathematical libraries. The classical problem of developing flow between parallel-plates is analysed first, as for both uniform and irrotational inlet conditions. The effect of truncating the duct length in the accuracy of the obtained solution is studied. A convergence analysis of the results obtained by the GITT is performed and compared with results obtained by finite difference and finite element methods, for different values of Reynolds number. The problem of flow over a backward-facing step then follows. Comparisons with experimental results in the literature indicate an excellent agreement. The numerical co-validation was established for a test case, and perfect agreement is reached against results considered as benchmarks in the recent literature. The results were shown to be physically more reasonable than others obtained by purely numerical methods, in particular for situations where three-dimensional effects are identified. Finally, a test problem for an irregular by shoped duct was studied and compared against results found in the literature, with good agreement and excellent convergence rates for the stream function field along the whole channel, for different values of Reynolds number. (author) 78 refs., 24 figs., 14 tabs.

  5. Asymptotic Convergence Encryption of Periodic Solutions for Euler-Poisson Equation%欧拉-泊松方程的周期解渐近性收敛加密

    屈哲; 陶可勤


    欧拉-泊松方程的椭圆函数周期解渐近性收敛模型是实现浮点数据模糊加密核心基础,广泛应用在通信编码和数据加密等领域。对浮点数据的模糊加密能有效保证网络中实时数据交互通信的安全,通过对浮点数据模糊加密稀疏集准确构造建模,提高加密性能。提出采用欧拉-泊松方程的椭圆函数周期解渐近性收敛数学建模的方法实现对浮点数据进行模糊加密,利用压缩映射原理来完成特征解分区处理,给出在控制单元的作用下对浮点数据进行密钥重整,求得欧拉-泊松方程椭圆函数在不定搜索下的三孤波解。通过数学推导证明了欧拉-泊松方程椭圆函数的周期解式渐进收敛性,实现对大数据库的数据加密,实验得出该加密数学模型的收敛性能较好,性能优越。%The elliptic function periodic solutions of the Euler Poisson equations asymptotic convergence of the model is the realization of floating-point data encryption based fuzzy core, widely used in the communication code and data encryption and other fields. The fuzzy of floating-point data encryption can effectively guarantee the real-time data communication net⁃work security, based on fuzzy set accurate floating-point data encryption sparse structure modeling, improve the encryption performance. Proposed elliptic function periodic using Euler Poisson equation method for the solution of asymptotic conver⁃gence of mathematical modeling to achieve floating-point fuzzy data encryption, by using the contraction mapping principle to complete the characteristic solution separately are given in the control unit under the action of the floating point data are key reforming, obtained the Euler Poisson equations of elliptic function in indefinite search the three solitary wave solu⁃tions. Through mathematical derivation of gradual convergence of the cycle solution of Euler Poisson equations of elliptic function is

  6. Remark on Global Existence of Smooth Solutions to Three-Dimensional Compressible Isentropic Euler Equations for Chaplygin Gases%三维可压等熵Euler方程光滑解的整体存在性

    赵娟; 王银霞


    In this paper we prove that global existence of smooth solutions to Cauchy problem for compressible isentropic Euler equations for Chaplygin gases or Karman-Tsien gases in three space dimensions , provided that the initial data are obtained by adding a small smooth perturbation with compact support to a constant state and the vorticity of the initial velocity vanishes.%研究了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.如果初值是一个常状态的小扰动并且初速度的旋度等于零,证明了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.

  7. Quasi-neutral limit of the full bipolar Euler-Poisson system


    The quasi-neutral limit of the multi-dimensional non-isentropic bipolar Euler-Poisson system is considered in the present paper. It is shown that for well-prepared initial data the smooth solution of the non-isentropic bipolar Euler-Poisson system converges strongly to the compressible non-isentropic Euler equations as the Debye length goes to zero.

  8. An Euler-Lagrange method considering bubble radial dynamics for modeling sonochemical reactors.

    Jamshidi, Rashid; Brenner, Gunther


    Unsteady numerical computations are performed to investigate the flow field, wave propagation and the structure of bubbles in sonochemical reactors. The turbulent flow field is simulated using a two-equation Reynolds-Averaged Navier-Stokes (RANS) model. The distribution of the acoustic pressure is solved based on the Helmholtz equation using a finite volume method (FVM). The radial dynamics of a single bubble are considered by applying the Keller-Miksis equation to consider the compressibility of the liquid to the first order of acoustical Mach number. To investigate the structure of bubbles, a one-way coupling Euler-Lagrange approach is used to simulate the bulk medium and the bubbles as the dispersed phase. Drag, gravity, buoyancy, added mass, volume change and first Bjerknes forces are considered and their orders of magnitude are compared. To verify the implemented numerical algorithms, results for one- and two-dimensional simplified test cases are compared with analytical solutions. The results show good agreement with experimental results for the relationship between the acoustic pressure amplitude and the volume fraction of the bubbles. The two-dimensional axi-symmetric results are in good agreement with experimentally observed structure of bubbles close to sonotrode.

  9. Interpolation by two-dimensional cubic convolution

    Shi, Jiazheng; Reichenbach, Stephen E.


    This paper presents results of image interpolation with an improved method for two-dimensional cubic convolution. Convolution with a piecewise cubic is one of the most popular methods for image reconstruction, but the traditional approach uses a separable two-dimensional convolution kernel that is based on a one-dimensional derivation. The traditional, separable method is sub-optimal for the usual case of non-separable images. The improved method in this paper implements the most general non-separable, two-dimensional, piecewise-cubic interpolator with constraints for symmetry, continuity, and smoothness. The improved method of two-dimensional cubic convolution has three parameters that can be tuned to yield maximal fidelity for specific scene ensembles characterized by autocorrelation or power-spectrum. This paper illustrates examples for several scene models (a circular disk of parametric size, a square pulse with parametric rotation, and a Markov random field with parametric spatial detail) and actual images -- presenting the optimal parameters and the resulting fidelity for each model. In these examples, improved two-dimensional cubic convolution is superior to several other popular small-kernel interpolation methods.

  10. On Euler's problem

    Egorov, Yurii V [Institute de Mathematique de Toulouse, Toulouse (France)


    We consider the classical problem on the tallest column which was posed by Euler in 1757. Bernoulli-Euler theory serves today as the basis for the design of high buildings. This problem is reduced to the problem of finding the potential for the Sturm-Liouville equation corresponding to the maximum of the first eigenvalue. The problem has been studied by many mathematicians but we give the first rigorous proof of the existence and uniqueness of the optimal column and we give new formulae which let us find it. Our method is based on a new approach consisting in the study of critical points of a related nonlinear functional. Bibliography: 6 titles.


    Wang, Yougang; Xu, Yidong; Chen, Xuelei [Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012 China (China); Park, Changbom [School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722 (Korea, Republic of); Kim, Juhan, E-mail:, E-mail: [Center for Advanced Computation, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722 (Korea, Republic of)


    We study the two-dimensional topology of the 21-cm differential brightness temperature for two hydrodynamic radiative transfer simulations and two semi-numerical models. In each model, we calculate the two-dimensional genus curve for the early, middle, and late epochs of reionization. It is found that the genus curve depends strongly on the ionized fraction of hydrogen in each model. The genus curves are significantly different for different reionization scenarios even when the ionized faction is the same. We find that the two-dimensional topology analysis method is a useful tool to constrain the reionization models. Our method can be applied to the future observations such as those of the Square Kilometre Array.

  12. Two dimensional topology of cosmological reionization

    Wang, Yougang; Xu, Yidong; Chen, Xuelei; Kim, Juhan


    We study the two-dimensional topology of the 21-cm differential brightness temperature for two hydrodynamic radiative transfer simulations and two semi-numerical models. In each model, we calculate the two dimensional genus curve for the early, middle and late epochs of reionization. It is found that the genus curve depends strongly on the ionized fraction of hydrogen in each model. The genus curves are significantly different for different reionization scenarios even when the ionized faction is the same. We find that the two-dimensional topology analysis method is a useful tool to constrain the reionization models. Our method can be applied to the future observations such as those of the Square Kilometer Array.

  13. The Characteristics Method Applied to Stationary Two-Dimensional and Rotationally Symmetrical Gas Flows

    Pfeiffer, F.; Meyer-Koenig, W.


    By means of characteristics theory, formulas for the numerical treatment of stationary compressible supersonic flows for the two-dimensional and rotationally symmetrical cases have been obtained from their differential equations.

  14. A novel schedule for solving the two-dimensional diffusion problem in fractal heat transfer

    Xu Shu


    Full Text Available In this work, the local fractional variational iteration method is employed to obtain approximate analytical solution of the two-dimensional diffusion equation in fractal heat transfer with help of local fractional derivative and integral operators.

  15. Two-dimensional x-ray diffraction

    He, Bob B


    Written by one of the pioneers of 2D X-Ray Diffraction, this useful guide covers the fundamentals, experimental methods and applications of two-dimensional x-ray diffraction, including geometry convention, x-ray source and optics, two-dimensional detectors, diffraction data interpretation, and configurations for various applications, such as phase identification, texture, stress, microstructure analysis, crystallinity, thin film analysis and combinatorial screening. Experimental examples in materials research, pharmaceuticals, and forensics are also given. This presents a key resource to resea

  16. Matching Two-dimensional Gel Electrophoresis' Spots

    Dos Anjos, António; AL-Tam, Faroq; Shahbazkia, Hamid Reza


    This paper describes an approach for matching Two-Dimensional Electrophoresis (2-DE) gels' spots, involving the use of image registration. The number of false positive matches produced by the proposed approach is small, when compared to academic and commercial state-of-the-art approaches. This ar......This paper describes an approach for matching Two-Dimensional Electrophoresis (2-DE) gels' spots, involving the use of image registration. The number of false positive matches produced by the proposed approach is small, when compared to academic and commercial state-of-the-art approaches...

  17. Towards two-dimensional search engines

    Ermann, Leonardo; Chepelianskii, Alexei D.; Shepelyansky, Dima L.


    We study the statistical properties of various directed networks using ranking of their nodes based on the dominant vectors of the Google matrix known as PageRank and CheiRank. On average PageRank orders nodes proportionally to a number of ingoing links, while CheiRank orders nodes proportionally to a number of outgoing links. In this way the ranking of nodes becomes two-dimensional that paves the way for development of two-dimensional search engines of new type. Statistical properties of inf...

  18. Energy Conservation in Two-dimensional Incompressible Ideal Fluids

    Cheskidov, A.; Filho, M. C. Lopes; Lopes, H. J. Nussenzveig; Shvydkoy, R.


    This note addresses the issue of energy conservation for the 2D Euler system with an L p -control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if {ω = nabla × u in L^{3/2}}. An example of a 2D field in the class {ω in L^{3/2 - ɛ}} for any ɛ > 0, and {u in B^{1/3}_{3,∞}} (Onsager critical space, see Shvydkoy in Discr Contin Dyn Syst Ser S 3(3):473-496, 2010) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument, which does not differentiate between 2D and 3D, and requires Onsager's regularity control on the solution. Next, we show that for physically realizable solutions there is a mechanism preventing the anomalous dissipation in 2D that does not require such a control. Namely, we prove that any solution to the Euler equations produced via a vanishing viscosity limit from the Navier-Stokes equations, with {ω in L^p}, for p > 1, conserves energy.

  19. Robust and Simple Non-Reflecting Boundary Conditions for the Euler Equations - A New Approach based on the Space-Time CE/SE Method

    Chang, S.-C.; Himansu, A.; Loh, C.-Y.; Wang, X.-Y.; Yu, S.-T.J.


    This paper reports on a significant advance in the area of nonreflecting boundary conditions (NRBCs) for unsteady flow computations. As a part of t he development of t he space-time conservation element and solution element (CE/SE) method, sets of NRBCs for 1D Euler problems are developed without using any characteristics- based techniques. These conditions are much simpler than those commonly reported in the literature, yet so robust that they are applicable to subsonic, transonic and supersonic flows even in the presence of discontinuities. In addition, the straightforward multidimensional extensions of the present 1D NRBCs have been shown numerically to be equally simple and robust. The paper details the theoretical underpinning of these NRBCs, and explains t heir unique robustness and accuracy in terms of t he conservation of space-time fluxes. Some numerical results for an extended Sod's shock-tube problem, illustrating the effectiveness of the present NRBCs are included, together with an associated simple Fortran computer program. As a preliminary to the present development, a review of the basic CE/SE schemes is also included.

  20. 一种基于高阶修正双曲线距离方程的中轨道SAR二维频谱%A Two-dimensional Spectrum for MEO SAR Based on High-order Modified Hyperbolic Range Equation

    包敏; 邢孟道; 李亚超; 保铮


    Because of the long integration time of Medium Earth Orbit SAR (MEO SAR), the hyperbolic range equation based on linear trajectory module is not suit for MEO SAR. Considering this issue, a high-order modified hyperbolic range equation is proposed. Incorporating with an additional linear component and quartic component, quartic Taylor series expansion of it has exactly the same value as which of the actual range history of MEO SAR. Then, the two-dimensional spectrum based on high-order modified hyperbolic range is analytically derived by using an approximate azimuth stationary point, based on method of series reversion the accuracy of the two-dimensional spectrum is analyzed which is exactly equal to quartic phase term. Simulation results show that the proposed range equation and the two-dimensional spectrum are accurate which can give fine resolution imagery with the entire aperture.%中轨道合成孔径雷达(MEO SAR)轨道高度高,合成孔径时间长,直线运动轨迹模型下的双曲线距离方程不再适用.针对这一问题,该文提出了一种适用于MEO SAR的高阶修正双曲线距离方程,该距离方程通过引入一线性项和一四次项对双曲线距离方程进行修正,使得其能对MEO SAR真实斜距历程进行四阶精确逼近.在此基础上,采用驻相点近似的方法推导该距离方程下2维频谱的闭合解析解,并结合级数反演法对频谱精度进行分析,发现采用驻相点近似方法得到的频谱精度严格精确到四次相位项,能满足MEO SAR精确成像的要求,为了便于成像算法的设计,该文对2维频谱各部分的空变性进行了分析.最后,仿真结果表明:该文距离方程和频谱精度较高,能实现MEO SAR全孔径精确成像.

  1. The Persistence Problem in Two-Dimensional Fluid Turbulence

    Perlekar, Prasad; Mitra, Dhrubaditya; Pandit, Rahul


    We present a natural framework for studying the persistence problem in two-dimensional fluid turbulence by using the Okubo-Weiss parameter {\\Lambda} to distinguish between vortical and extensional regions. We then use a direct numerical simulation (DNS) of the two-dimensional, incompressible Navier-Stokes equation with Ekman friction to study probability distribution functions (PDFs) of the persistence times of vortical and extensional regions by employing both Eulerian and Lagrangian measurements. We find that, in the Eulerian case, the persistence-time PDFs have exponential tails; by contrast, this PDF for Lagrangian particles, in vortical regions, has a power-law tail with a universal exponent {\\theta} = 3.1 \\pm 0.2.

  2. Two Dimensional Lattice Boltzmann Method for Cavity Flow Simulation

    Panjit MUSIK


    Full Text Available This paper presents a simulation of incompressible viscous flow within a two-dimensional square cavity. The objective is to develop a method originated from Lattice Gas (cellular Automata (LGA, which utilises discrete lattice as well as discrete time and can be parallelised easily. Lattice Boltzmann Method (LBM, known as discrete Lattice kinetics which provide an alternative for solving the Navier–Stokes equations and are generally used for fluid simulation, is chosen for the study. A specific two-dimensional nine-velocity square Lattice model (D2Q9 Model is used in the simulation with the velocity at the top of the cavity kept fixed. LBM is an efficient method for reproducing the dynamics of cavity flow and the results which are comparable to those of previous work.

  3. Two-dimensional localized structures in harmonically forced oscillatory systems

    Ma, Y.-P.; Knobloch, E.


    Two-dimensional spatially localized structures in the complex Ginzburg-Landau equation with 1:1 resonance are studied near the simultaneous presence of a steady front between two spatially homogeneous equilibria and a supercritical Turing bifurcation on one of them. The bifurcation structures of steady circular fronts and localized target patterns are computed in the Turing-stable and Turing-unstable regimes. In particular, localized target patterns grow along the solution branch via ring insertion at the core in a process reminiscent of defect-mediated snaking in one spatial dimension. Stability of axisymmetric solutions on these branches with respect to axisymmetric and nonaxisymmetric perturbations is determined, and parameter regimes with stable axisymmetric oscillons are identified. Direct numerical simulations reveal novel depinning dynamics of localized target patterns in the radial direction, and of circular and planar localized hexagonal patterns in the fully two-dimensional system.

  4. Enstrophy inertial range dynamics in generalized two-dimensional turbulence

    Iwayama, Takahiro; Watanabe, Takeshi


    We show that the transition to a k-1 spectrum in the enstrophy inertial range of generalized two-dimensional turbulence can be derived analytically using the eddy damped quasinormal Markovianized (EDQNM) closure. The governing equation for the generalized two-dimensional fluid system includes a nonlinear term with a real parameter α . This parameter controls the relationship between the stream function and generalized vorticity and the nonlocality of the dynamics. An asymptotic analysis accounting for the overwhelming dominance of nonlocal triads allows the k-1 spectrum to be derived based upon a scaling analysis. We thereby provide a detailed analytical explanation for the scaling transition that occurs in the enstrophy inertial range at α =2 in terms of the spectral dynamics of the EDQNM closure, which extends and enhances the usual phenomenological explanations.

  5. Piezoelectricity in Two-Dimensional Materials

    Wu, Tao


    Powering up 2D materials: Recent experimental studies confirmed the existence of piezoelectricity - the conversion of mechanical stress into electricity - in two-dimensional single-layer MoS2 nanosheets. The results represent a milestone towards embedding low-dimensional materials into future disruptive technologies. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA.

  6. Kronecker Product of Two-dimensional Arrays

    Lei Hu


    Kronecker sequences constructed from short sequences are good sequences for spread spectrum communication systems. In this paper we study a similar problem for two-dimensional arrays, and we determine the linear complexity of the Kronecker product of two arrays. Our result shows that similar good property on linear complexity holds for Kronecker product of arrays.

  7. A novel two dimensional particle velocity sensor

    Pjetri, Olti; Wiegerink, Remco J.; Lammerink, Theo S.; Krijnen, Gijs J.


    In this paper we present a two wire, two-dimensional particle velocity sensor. The miniature sensor of size 1.0x2.5x0.525 mm, consisting of only two crossed wires, shows excellent directional sensitivity in both directions, thus requiring no directivity calibration, and is relatively easy to fabrica

  8. Two-dimensional microstrip detector for neutrons

    Oed, A. [Institut Max von Laue - Paul Langevin (ILL), 38 - Grenoble (France)


    Because of their robust design, gas microstrip detectors, which were developed at ILL, can be assembled relatively quickly, provided the prefabricated components are available. At the beginning of 1996, orders were received for the construction of three two-dimensional neutron detectors. These detectors have been completed. The detectors are outlined below. (author). 2 refs.

  9. Two-dimensional magma-repository interactions

    Bokhove, O.


    Two-dimensional simulations of magma-repository interactions reveal that the three phases --a shock tube, shock reflection and amplification, and shock attenuation and decay phase-- in a one-dimensional flow tube model have a precursor. This newly identified phase ``zero'' consists of the impact of

  10. Two-dimensional subwavelength plasmonic lattice solitons

    Ye, F; Hu, B; Panoiu, N C


    We present a theoretical study of plasmonic lattice solitons (PLSs) formed in two-dimensional (2D) arrays of metallic nanowires embedded into a nonlinear medium with Kerr nonlinearity. We analyze two classes of 2D PLSs families, namely, fundamental and vortical PLSs in both focusing and defocusing media. Their existence, stability, and subwavelength spatial confinement are studied in detai

  11. A two-dimensional Dirac fermion microscope

    Bøggild, Peter; Caridad, Jose; Stampfer, Christoph


    in the solid state. Here we provide a perspective view on how a two-dimensional (2D) Dirac fermion-based microscope can be realistically implemented and operated, using graphene as a vacuum chamber for ballistic electrons. We use semiclassical simulations to propose concrete architectures and design rules of 2...

  12. Duffing振子中随机微分方程的"欧拉-丸山"数值解法%Euler-maruyama method for stochastic differential equations of Duffing oscillator

    张嵩; 芮国胜; 孙文俊; 张洋; 崔文


    The moment equations of Duffing oscillator in the presence of noise and weak periodic signal are typical stochastic differential equations (SDE). Because of the presence noise,it is impossible to get the exact solutions to these equations. Simulink model and Runge-Kutta algorithm are usually used to solute SDE. In this paper, how the Euler Maruyama method can be used to numerically simulate the SDE of Duffing oscillator is studied. The numerical results show that Euler-Maruyama method can effectively simulate Duffing oscillator. Especially,the simulation elapsed time of this method is much shorter than that of the usually used methods.%引人噪声时的Duffing方程是一个二阶非线性随机微分方程,由于无法求得该随机微分方程的精确解析解,所以绝大多数情况下利用Simulink仿真模型,采用龙格库塔算法求Duffing方程的数值解.针对龙格库塔算法在每一步积分迭代过程中计算量大,导致系统进入稳态解的时间过长的问题,通过对布朗运动及其积分的分析,利用"欧拉-丸山"算法求解Duffing系统中的随机微分方程.经仿真计算,欧拉-丸山算法可以正确描述Duffing方程丰富的混沌动力学特性,而且在求解过程中能方便地定义随机积分方程中的噪声信号.仿真结果表明与常用的龙格库塔方法相比,"欧拉-丸山"算法具有较为明显的时间复杂度优势.

  13. Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces

    Bacigalupo, Andrea; Gambarotta, Luigi


    Dispersive waves in two-dimensional blocky materials with periodic microstructure made up of equal rigid units, having polygonal centro-symmetric shape with mass and gyroscopic inertia, connected with each other through homogeneous linear interfaces, have been analyzed. The acoustic behavior of the resulting discrete Lagrangian model has been obtained through a Floquet-Bloch approach. From the resulting eigenproblem derived by the Euler-Lagrange equations for harmonic wave propagation, two acoustic branches and an optical branch are obtained in the frequency spectrum. A micropolar continuum model to approximate the Lagrangian model has been derived based on a second-order Taylor expansion of the generalized macro-displacement field. The constitutive equations of the equivalent micropolar continuum have been obtained, with the peculiarity that the positive definiteness of the second-order symmetric tensor associated to the curvature vector is not guaranteed and depends both on the ratio between the local tangent and normal stiffness and on the block shape. The same results have been obtained through an extended Hamiltonian derivation of the equations of motion for the equivalent continuum that is related to the Hill-Mandel macro homogeneity condition. Moreover, it is shown that the hermitian matrix governing the eigenproblem of harmonic wave propagation in the micropolar model is exact up to the second order in the norm of the wave vector with respect to the same matrix from the discrete model. To appreciate the acoustic behavior of some relevant blocky materials and to understand the reliability and the validity limits of the micropolar continuum model, some blocky patterns have been analyzed: rhombic and hexagonal assemblages and running bond masonry. From the results obtained in the examples, the obtained micropolar model turns out to be particularly accurate to describe dispersive functions for wavelengths greater than 3-4 times the characteristic dimension of



    An approach of modeling viscosity, unsteady partially cavitating flows around lifting bodies is presented. By employing an one-fluid Navier-Stokers solver, the algorithm is proved to be able to handle two-dimensional laminar cavitating flows at moderate Reynolds number. Based on the state equation of water-vapor mixture, the constructive relations of densities and pressures are established. To numerically simulate the cavity wall, different pseudo transition of density models are presumed. The finite-volume method is adopted and the algorithm can be extended to three-dimensional cavitating flows.

  15. Electronics based on two-dimensional materials.

    Fiori, Gianluca; Bonaccorso, Francesco; Iannaccone, Giuseppe; Palacios, Tomás; Neumaier, Daniel; Seabaugh, Alan; Banerjee, Sanjay K; Colombo, Luigi


    The compelling demand for higher performance and lower power consumption in electronic systems is the main driving force of the electronics industry's quest for devices and/or architectures based on new materials. Here, we provide a review of electronic devices based on two-dimensional materials, outlining their potential as a technological option beyond scaled complementary metal-oxide-semiconductor switches. We focus on the performance limits and advantages of these materials and associated technologies, when exploited for both digital and analog applications, focusing on the main figures of merit needed to meet industry requirements. We also discuss the use of two-dimensional materials as an enabling factor for flexible electronics and provide our perspectives on future developments.

  16. Two-dimensional ranking of Wikipedia articles

    Zhirov, A. O.; Zhirov, O. V.; Shepelyansky, D. L.


    The Library of Babel, described by Jorge Luis Borges, stores an enormous amount of information. The Library exists ab aeterno. Wikipedia, a free online encyclopaedia, becomes a modern analogue of such a Library. Information retrieval and ranking of Wikipedia articles become the challenge of modern society. While PageRank highlights very well known nodes with many ingoing links, CheiRank highlights very communicative nodes with many outgoing links. In this way the ranking becomes two-dimensional. Using CheiRank and PageRank we analyze the properties of two-dimensional ranking of all Wikipedia English articles and show that it gives their reliable classification with rich and nontrivial features. Detailed studies are done for countries, universities, personalities, physicists, chess players, Dow-Jones companies and other categories.

  17. Two-Dimensional NMR Lineshape Analysis

    Waudby, Christopher A.; Ramos, Andres; Cabrita, Lisa D.; Christodoulou, John


    NMR titration experiments are a rich source of structural, mechanistic, thermodynamic and kinetic information on biomolecular interactions, which can be extracted through the quantitative analysis of resonance lineshapes. However, applications of such analyses are frequently limited by peak overlap inherent to complex biomolecular systems. Moreover, systematic errors may arise due to the analysis of two-dimensional data using theoretical frameworks developed for one-dimensional experiments. Here we introduce a more accurate and convenient method for the analysis of such data, based on the direct quantum mechanical simulation and fitting of entire two-dimensional experiments, which we implement in a new software tool, TITAN (TITration ANalysis). We expect the approach, which we demonstrate for a variety of protein-protein and protein-ligand interactions, to be particularly useful in providing information on multi-step or multi-component interactions.

  18. Towards two-dimensional search engines

    Ermann, Leonardo; Shepelyansky, Dima L


    We study the statistical properties of various directed networks using ranking of their nodes based on the dominant vectors of the Google matrix known as PageRank and CheiRank. On average PageRank orders nodes proportionally to a number of ingoing links, while CheiRank orders nodes proportionally to a number of outgoing links. In this way the ranking of nodes becomes two-dimensional that paves the way for development of two-dimensional search engines of new type. Information flow properties on PageRank-CheiRank plane are analyzed for networks of British, French and Italian Universities, Wikipedia, Linux Kernel, gene regulation and other networks. Methods of spam links control are also analyzed.

  19. Toward two-dimensional search engines

    Ermann, L.; Chepelianskii, A. D.; Shepelyansky, D. L.


    We study the statistical properties of various directed networks using ranking of their nodes based on the dominant vectors of the Google matrix known as PageRank and CheiRank. On average PageRank orders nodes proportionally to a number of ingoing links, while CheiRank orders nodes proportionally to a number of outgoing links. In this way, the ranking of nodes becomes two dimensional which paves the way for the development of two-dimensional search engines of a new type. Statistical properties of information flow on the PageRank-CheiRank plane are analyzed for networks of British, French and Italian universities, Wikipedia, Linux Kernel, gene regulation and other networks. A special emphasis is done for British universities networks using the large database publicly available in the UK. Methods of spam links control are also analyzed.

  20. A two-dimensional Dirac fermion microscope

    Bøggild, Peter; Caridad, José M.; Stampfer, Christoph; Calogero, Gaetano; Papior, Nick Rübner; Brandbyge, Mads


    The electron microscope has been a powerful, highly versatile workhorse in the fields of material and surface science, micro and nanotechnology, biology and geology, for nearly 80 years. The advent of two-dimensional materials opens new possibilities for realizing an analogy to electron microscopy in the solid state. Here we provide a perspective view on how a two-dimensional (2D) Dirac fermion-based microscope can be realistically implemented and operated, using graphene as a vacuum chamber for ballistic electrons. We use semiclassical simulations to propose concrete architectures and design rules of 2D electron guns, deflectors, tunable lenses and various detectors. The simulations show how simple objects can be imaged with well-controlled and collimated in-plane beams consisting of relativistic charge carriers. Finally, we discuss the potential of such microscopes for investigating edges, terminations and defects, as well as interfaces, including external nanoscale structures such as adsorbed molecules, nanoparticles or quantum dots.