A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation

E Momoniat

doi:10.1155/2014/754543

Abstract

Two nonstandard finite difference schemes are derived to solve the regularized long wave equation. The criteria for choosing the “best” nonstandard approximation to the nonlinear term in the regularized long wave equation come from considering the modified equation. The two “best” nonstandard numerical schemes are shown to preserve conserved quantities when compared to an implicit scheme in which the nonlinear term is approximated in the usual way. Comparisons to the single solitary wave solution show significantly better results, measured in theL2andL∞norms, when compared to results obtained using a Petrov-Galerkin finite element method and a splitted quadratic B-spline collocation method. The growth in the error when simulating the single solitary wave solution using the two “best” nonstandard numerical schemes is shown to be linear implying the nonstandard finite difference schemes are conservative. The formation of an undular bore for both steep and shallow initial profiles is captured without the formation of numerical instabilities.

Highlights

In this paper we derive two nonstandard finite difference schemes to solve the regularized long wave (RLW) equation
In this paper we have shown the following: (i) that the modified equation can be used as a criterion for selecting the “best” nonstandard finite difference scheme for solving the RLW equation (1); (ii) that the nonstandard numerical scheme we have chosen does perform well by preserving the three conserved quantities (5)–(7) and capturing the single solitary wave solution (8) accurately; (iii) that the nonstandard finite difference scheme captures the development of an undular bore for both steep and shallow initial profiles
Avilez-Valente and Seabra-Santos [6] summarize the computational efficiency of their method by indicating that their FEM method leads to a tridiagonal system of n + 1 equations, where n is the number of spatial finite elements

Summary

Momoniat

Two nonstandard finite difference schemes are derived to solve the regularized long wave equation. The criteria for choosing the “best” nonstandard approximation to the nonlinear term in the regularized long wave equation come from considering the modified equation. The two “best” nonstandard numerical schemes are shown to preserve conserved quantities when compared to an implicit scheme in which the nonlinear term is approximated in the usual way. Comparisons to the single solitary wave solution show significantly better results, measured in the L2 and L∞ norms, when compared to results obtained using a Petrov-Galerkin finite element method and a splitted quadratic B-spline collocation method. The growth in the error when simulating the single solitary wave solution using the two “best” nonstandard numerical schemes is shown to be linear implying the nonstandard finite difference schemes are conservative.

Introduction

Analysis of Modified Equation

Simulation of Numerical Schemes

Concluding Remarks