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

Abstract

An initial-boundary value problem for the n-dimensional wave equation is considered. A threelevel 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 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.