The Qualitative Behavior of the Solution of the Wave Equation using the Non-standard Finite Difference and Galerkin Methods

The optimal rate of convergence of the wave equation using the continuous Galerkin method in both the energy as well as the L-2-norms is well known. Here, we exploit the technique used above and design a fully-discrete scheme consisting of coupling the Non-standard finite difference method in time and the continuous Galerkin method in the space variable. We show that, for sufficiently smooth solution, the maximal error in these norms possesses the same order of convergence O(h(2) +(Delta t)(2)), where h is the mesh size. Furthermore, we show that the above scheme replicates the properties of the exact solution.