A New Fast Numerical Method Based on Off-Step Discretization for Two-Dimensional Quasilinear Hyperbolic Partial Differential Equations

Abstract

We report a new 3-level implicit compact numerical method of order four in time and four in space based on off-step discretization for the solution of two-space-dimensional quasilinear hyperbolic equation [Formula: see text], [Formula: see text], [Formula: see text] defined in the region [Formula: see text], [Formula: see text], [Formula: see text]. We require only 19 grid points for the unknown variable [Formula: see text] and two extra off-step points each in [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to two-dimensional hyperbolic equations with singular coefficients, which is the main attraction of our work. We do not require any fictitious points for computation. The proposed method when applied to a two-dimensional damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve damped wave equation. Many benchmark problems are solved to confirm the fourth-order convergence of the proposed method.