Multiscale Partially Explicit Splitting with Mass Lumping for High-Contrast Wave Equations

In this paper, contrast-independent partially explicit time discretization for wave equations in heterogeneous high-contrast media via mass lumping is concerned. By employing a mass lumping scheme to diagonalize the mass matrix, the matrix inversion procedures can be avoided, thereby significantly enhancing computational efficiency especially in the explicit part. In addition, after decoupling the resulting system, higher order time discretization techniques can be applied to get better accuracy within the same time step size. Furthermore, the spatial space is divided into two components: contrast-dependent ("fast") and contrast-independent ("slow") subspaces. Using this decomposition, our objective is to introduce an appropriate time splitting method that ensures stability and guarantees contrast-independent discretization under suitable conditions. We analyze the stability and convergence of the proposed algorithm. In particular, we discuss the second order central difference and higher order Runge-Kutta method for a wave equation. Several numerical examples are presented to confirm our theoretical results and to demonstrate that our proposed algorithm achieves high accuracy while reducing computational costs for high-contrast problems.