Multi-time-stepping splitting scheme with mass lumping for the multiscale wave equation

Abstract

Efficient simulation of wave propagation in high-contrast media is challenging because standard explicit schemes are subject to severe time-step restrictions induced by fine spatial scales and fast multiscale modes, whereas fully implicit schemes require costly globally coupled solves. In this work, we develop a mass-lumped multi-time-stepping partially explicit splitting method for multiscale wave equations. Based on a nonlocal multicontinuum (NLMC) decomposition of the approximation space, the solution is separated into contrast-dependent fast components and contrast-independent slow components, which are advanced implicitly and explicitly, respectively. A mass-lumping approximation diagonalizes the coarse-scale mass operator and removes the coupling between two subproblems, thus enabling multirate time integration for the two parts. To further improve efficiency, we introduce a residual-based adaptive strategy that dynamically adjusts the time step of the explicitly treated subsystem according to its temporal error indicator. The proposed scheme is shown to be energy stable under suitable assumptions. Numerical experiments demonstrate that the proposed mass-lumped multi-time-stepping scheme substantially reduces computational cost while maintaining accuracy comparable to that of standard fully implicit methods.