High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients

In this paper, we develop implicit difference schemes of O(k(4) + k(2)h(2) + h(4)), where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions. The proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable ADI method of O(k(4) + k(2)h(2) + h(4)). Some physical nonlinear problems are provided to demonstrate the accuracy of the implementation.