Contrast-independent partially explicit time discretizations for multiscale wave problems

Abstract: In this work, we design and investigate contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. We consider multiscale problems, where the spatial heterogeneities are at subgrid level and are not resolved. In our previous work, we have introduced contrast-independent partially explicit time discretizations and applied to parabolic equations. The main idea of contrast-independent partially explicit time discretization is to split the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Using this decomposition, our goal is further appropriately to introduce time splitting such that the resulting scheme is stable and can guarantee contrast-independent discretization under some suitable (reasonable) conditions. In this paper, we propose contrast-independent partially explicitly scheme for wave equations. The splitting requires a careful design. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (slow). This condition requires some type of non-contrast dependent space and is easier to satisfy in the "slow" space. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.