Optimal Rate of Convergence for a Nonstandard Finite Difference Galerkin Method Applied to Wave Equation Problems

Pius W M Chin

doi:10.1155/2013/520219

Abstract

The optimal rate of convergence of the wave equation in both the energy and theL2-norms using continuous Galerkin method is well known. We exploit this technique and design a fully discrete scheme consisting of coupling the nonstandard finite difference method in the time and the continuous Galerkin method in the space variables. We show that, for sufficiently smooth solution, the maximal error in theL2-norm possesses the optimal rate of convergenceO(h2+(Δt)2)wherehis the mesh size andΔtis the time step size. Furthermore, we show that this scheme replicates the properties of the exact solution of the wave equation. Some numerical experiments should be performed to support our theoretical analysis.

Highlights

Most physical phenomena such as the acoustics, electromagnetic, and elastic problems are modeled by the wave equation
Instead of the continuous Galerkin method summarized previously, we present a reliable scheme NSFDCG consisting of the nonstandard finite difference method in time and the continuous Galerkin method in the space variable
We show that the numerical solution obtained from the scheme NSFD-CG attained the optimal convergence in both the energy and L2-norms

Summary

Introduction

Most physical phenomena such as the acoustics, electromagnetic, and elastic problems are modeled by the wave equation. The methods which have been heavily used for the study of the wave equation (1)–(4) in physical life are the continuous as well as the discontinuous Galerkin methods; see [1, 2] for more details. We exploit and present a reliable technique consisting of coupling the nonstandard finite difference (NSFD) method in time and the continuous Galerkin (CG) method in the space variables. The technique is geared toward obtaining a sufficiently smooth solution, the maximal error in the L2-norm, and to show that the error across the entire interval convergences optimally as O(h2 + Δt2) where h is the mesh size and Δt is the time step size The reliability of this technique comes from the fact that the NSFD-CG method preserves both the energy features and the hyperbolicity of the exact solution of the wave equation (1)– (4).

Notations

The Continuous Galerkin Method

Coupled Nonstandard Finite Difference and Continuous Galerkin Methods

Numerical Experiments

Conclusion