A New Fast Algorithm Based on Half-Step Discretization for 3D Quasilinear Hyperbolic Partial Differential Equations

Abstract

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.