On properties of an explicit in time fourth-order vector compact scheme for the multidimensional wave equation

TL;DR

This paper extends a 4th-order explicit compact scheme for the multidimensional wave equation to arbitrary dimensions and meshes, providing stability, energy conservation, and error bounds, along with generalizations to variable coefficients.

Contribution

It generalizes a 4th-order explicit scheme from 2D to n-dimensional problems and introduces new stability and energy conservation results.

Findings

New stability bounds in mesh energy norms

Discrete energy conservation laws established

Rigorous proof of 4th order error bound

Abstract

An initial-boundary value problem for the $n$-dimensional wave equation is considered. A three-level explicit in time and conditionally stable 4th-order compact scheme constructed recently for $n=2$ and the square mesh is generalized to the case of any $n\geq 1$ and the rectangular uniform mesh. Another approach to approximate the solution at the first time level (not exploiting high-order derivatives of the initial functions) is suggested. New stability bounds in the mesh energy norms and the discrete energy conservation laws are given, and the 4th order error bound is rigorously proved. Generalizations to the cases of the non-uniform meshes in space and time as well as of the wave equation with variable coefficients are suggested.