An operator splitting technique for an unconditionally stable difference method for a linear three space dimensional hyperbolic equation with variable coefficients

We report a new three-step operator splitting method of O(k2 + h2) for the difference solution of linear hyperbolic equation uu + 2α(x,y,z,t)ut + β2(x,y,z,t)u = A(x,y, z,t)uxx + B(x,y,z,t)uyy + C(x,y,z,t)uzz + f(x,y,z,t) subject to appropriate initial and Dirichlet boundary conditions, where α(x,y,z,t) > β(x,y,z,t) > 0 and A(x,y,z,t) > 0, B(x,y,z,t) > 0, C(x,y,z,t) > 0. The method is applicable to singular problems and stable for all choices of h > 0 and k > 0. The resulting system of algebraic equations is solved by using a tri-diagonal solver. Computational results are provided to demonstrate the viability of the new method.