An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients

We propose a three level implicit unconditionally stable difference scheme of O(k(2) + h(2)) for the difference solution of second order linear hyperbolic equation u(n) + 2 alpha(x, t)u(t) + beta(2)(x, t)u = A(x, t)u(xx) + f(x, t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where A(x, t) > 0, alpha(x, t) > beta(x, t) >= 0. The proposed formula is applicable to the problems having singularity at x = 0. The resulting tri-diagonal linear system of equations is solved by using Gauss-elimination method. Numerical examples are provided to illustrate the unconditionally stable character of the proposed method. (c) 2004 Elsevier Inc. All rights reserved.