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(k2+h2) for the difference solution of second order linear hyperbolic equation utt + 2α(x,t)ut + β2(x,t)u = A(x,t)uxx + f(x,t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where A(x, t) > 0, a(x, t) > β(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.