An Alternating Direction Implicit Finite Difference Method For Second Order Hyperbolic Equations

Abstract

In recent years, much attention has been given in the literature to the development, analysis, and implementation of finite difference schemes for the numerical solution of hyperbolic equations, which can describe physical phenomenons of vibrating string, elastic film and three-dimensional elastomers. In most of the cases, we use explicit difference schemes or implicit difference schemes to find numerical solutions of hyperbolic equations. The formers are suitable for parallel computation but it has limitations of stability. The latters are generally stable, but it is necessary to solve different linear systems at each level of time, which leads to more computational cost and time. In this paper, a two-level Crank-Nicolson alternating direction implicit (ADI) difference scheme is derived for solving the second order hyperbolic equations with variable coefficients by introducing the auxiliary variable. Convergence and stability analysis of the ADI scheme are given by the energy method. Finally, numerical examples are presented to illustrate the efficiency of the ADI difference scheme. Keywords-implicit difference; alternating-direction; variable substitution; convergence; stability