A New High-Order Approximation for the Solution of Two-Space-Dimensional Quasilinear Hyperbolic Equations

we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form utt A x, y, t, u uxx B x, y, t, u uyy g x, y, t, u, ux, uy, ut , 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions , where k > 0 and h > 0 are mesh sizes in time and space directions, respectively. We use only five evaluations of the function g as compared to seven evaluations of the same function discussed by Mohanty et al., 1996 and 2001 . We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Some examples and their numerical results are provided to justify the usefulness of the proposed method.