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 utt+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.