High order variable mesh off-step discretization for the solution of 1-D non-linear hyperbolic equation

In this paper, we propose a new high order three-level implicit method based on off-step discretization on a non-uniform mesh for the solution of 1-D non-linear hyperbolic partial differential equation of the form utt = uxx + g(x, t, u, ux, ut), subject to appropriate initial and Dirichlet boundary conditions. We use only three evaluations of the function g and three grid points at each time level in a compact cell. Our method is directly applicable to the wave equation in polar coordinates and we do not require any special technique to handle singular coefficients of the differential equation. The method is convergent for uniform mesh. Numerical results are provided to justify the usefulness of the proposed method.