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, α(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.