Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

Abstract

We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear Klein-Gordon equation with a triangle discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.

On a weakly L‐stable time integration formula coupled with nonstandard finite difference scheme with application to nonlinear parabolic partial differential equations

“…To discretize the second‐order spatial derivative, we use the NSFD scheme 43 . The following are the main advantages of using the NSFD scheme 44–46 : The accuracy, positivity, boundedness, and monotonicity of the solutions are better with the NSFD scheme compared to the standard finite difference scheme. It does not allow to occur the numerical instabilities, whereas other methods may lose their stability. Depending on the nature of the problems, this method allows us to select a different denominator function. It preserves certain properties of the exact solutions of the differential equations and shows a good convergence behavior. …”

An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation

“…A nonstandard finite difference scheme can be constructed from the exact finite difference scheme [4]. An exact finite difference scheme can be constructed for any ordinary differential equation (ODE) or partial differential equation (PDE) from the analytical solution of the differential equation [5][6][7]. Among the various numerical techniques such as classical finite difference, finite volume, adaptive mesh, finite element, and spectral method for solving ODEs and PDEs, NSFD schemes have been proved to be one of the most efficient approaches in recent years.…”

A Numerical Comparison for a Discrete HIV Infection of CD4+T-Cell Model Derived from Nonstandard Numerical Scheme

“…These schemes are developed for compensating the weaknesses that may be caused by standard difference methods, for example, numerical instabilities. Also, the dynamic consistency could be presented well by NFDS [7]. The most important advantage of this scheme is that, choosing a convenient denominator function instead of the step size ℎ, better results can be obtained.…”