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

Abstract

The optimal rate of convergence of the wave equation in both the energy and the $L^2$-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 the $L^2$-norm possesses the optimal rate of convergence $O(h^2+{(\mathrm{\Delta }t)}^2)$ where $h$ is the mesh size and $\mathrm{\Delta }t$ is 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.