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

Canan Koroglu,Ayhan Aydin

doi:10.1155/2017/4796070

Abstract

A numerical solution of the modified Korteweg-de Vries (MKdV) equation is presented by using a nonstandard finite difference (NSFD) scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization. Local truncation error of the NSFD scheme and linear stability analysis are discussed. To test the accuracy and efficiency of the method, some numerical examples are given. The numerical results of NSFD scheme are compared with the exact solution and a standard finite difference scheme. The numerical results illustrate that the NSFD scheme is a robust numerical tool for the numerical integration of the MKdV equation.

Highlights

This paper is concerned with the nonstandard integration of modified Korteweg-de Vries (MKdV) equation ut + qu2ux + ruxxx = 0, (x, t) ∈ [xL, xR] × [0, T] (1) with initial condition u (x, 0) = u0 (x), x ∈ [xL, xR]
The authors in [8] proposed a nonstandard finite volume method for the numerical solution of a singularly perturbed Schrodinger equation. They have shown that the proposed nonstandard finite volume method is capable of reducing the computational cost associated with most classical schemes. They have highlighted that nonstandard finite difference (NSFD) schemes have been efficient in tackling the deficiency of classical finite difference scheme for the approximation of solutions of several differential equation models
It is well known that a NSFD method is constructed from the exact finite difference schemes

Summary

Introduction

It plays an important role in the study of nonlinear physics such as fluid physics and quantum field theory It is a model equation for the weakly nonlinear long waves which occur in many different physical systems. The authors in [8] proposed a nonstandard finite volume method for the numerical solution of a singularly perturbed Schrodinger equation They have shown that the proposed nonstandard finite volume method is capable of reducing the computational cost associated with most classical schemes. They have highlighted that NSFD schemes have been efficient in tackling the deficiency of classical finite difference scheme for the approximation of solutions of several differential equation models.

Nonstandard Discretization

Stability and Local Truncation Error

Numerical Results

Conclusion