On higher-order compact ADI schemes for the variable coefficient wave equation

Abstract

We consider an initial-boundary value problem for the $n$-dimensional wave equation, $n\geq 2$, with the variable sound speed with the nonhomogeneous Dirichlet boundary conditions. We construct and study three-level in time and compact in space three-point in each spatial direction alternating direction implicit (ADI) schemes having the approximation orders $\mathcal{O}(h_t^2+|h|^4)$ and $\mathcal{O}(h_t^4+|h|^4)$ on the uniform rectangular mesh. The study includes stability bounds in the strong and weak energy norms, the discrete energy conservation law and the error bound of the order $\mathcal{O}(h_t^2+|h|^4)$ for the first scheme as well as a short justification of the approximation order $\mathcal{O}(h_t^4+|h|^4)$ for the second scheme. We also present results of numerical experiments.