High accuracy cubic spline approximation on a geometric mesh for the solution of 1D non-linear wave equations

Abstract

In this paper, we propose a new high order  three-level implicit compact discretization based on cubic spline approximation on a non-uniform mesh in space-direction for the solution of 1D non-linear hyperbolic partial differential equation of the form $u_{tt} = u_{tt} + G (x,t,u,u_x, u_t)$ subject to appropriate initial and Dirichlet boundary conditions. We use only three grid points at each time level and describe the derivation procedure in details. We also show how our method is able to handle the wave equation in polar coordinates. Numerical results are provided to justify the usefulness of the proposed method.