On stability and error bounds of an explicit in time higher-order vector compact scheme for the multidimensional wave and acoustic wave equations

Abstract

We study a three-level explicit in time higher-order vector compact scheme, with additional n sought functions approximating second order non-mixed spatial derivatives of the solution, for an initial-boundary value problem for the n-dimensional wave equation and acoustic wave equation, with the variable speed of sound, n⩾1. We also approximate the solution at the first time level in a similar two-level manner, without using derivatives of the initial data as usual. For the first time, under the CFL-type conditions, new stability bounds in the standard and stronger mesh energy norms and the discrete energy conservation laws are presented, and the corresponding error bounds of the orders 4 and 3.5 are rigorously proved. Generalizations to the cases of the nonuniform meshes in space and time are described. Results of various numerical experiments are also included.