A new three-level implicit cubic spline method for the solution of 1D quasi-linear hyperbolic equations

Abstract

In this paper, we study a new three-level implicit compact finite-difference discretization of O(k2 + k2h2 + h4), based on cubic spline approximation, for the solution of one-space dimensional second-order quasi-linear hyperbolic partial differential equations, where k > 0 and h > 0 are mesh sizes in time and space directions, respectively. We describe the complete derivation procedure of the method in detail and also discuss how our discretization is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear telegraphic equation is also shown to be unconditionally stable. Some examples and their numerical results are provided to justify the usefulness of the proposed method. © 2013 Springer Science+Business Media New York.