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

TL;DR

The paper tackles efficient numerical simulation of high-contrast wave equations by introducing a contrast-independent, partially explicit time discretization that leverages mass lumping to diagonalize the mass matrix $M$ into a diagonal $\tilde{M}$ and decouple fast/slow dynamics. It combines a two-scale spatial splitting with multiscale spaces constructed via auxiliary spaces $V_{\text{aux},1}$ and $V_{\text{aux},2}$, and global spaces $V_{\text{glo},1}$ and $V_{\text{glo},2}$ within a CEM-GMsFEM framework, followed by mass lumping to enable efficient decoupled time stepping. A four-stage, third-order partially explicit Runge-Kutta scheme is developed to advance the decoupled system, and stability and convergence are proven to be contrast-independent under reasonable conditions. Numerical experiments on three heterogeneous media demonstrate that the method achieves high accuracy with substantially reduced computational cost, with convergence rates governed by the time step $\tau$, coarse-grid size $H$, and oversampling, while mass lumping introduces negligible accuracy loss. The work provides a practical, scalable approach for simulating wave propagation in complex multiscale, high-contrast media.

Abstract

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.

Figures (11)

• Figure 1: Illustration of the fine grid, coarse grid $K_i$ and oversampling domain $K_{i,1}$.
• Figure 2: Left: the source term $f$. Right: the source term $f_2$.
• Figure 3: Upper left: the first media $\kappa_1$. Upper right: reference solution for Case 1. Lower left: second order central difference solution for Case 1. Lower right: third order Runge-Kutta solution for Case 1.
• Figure 4: Left: solution via Runge-Kutta scheme without mass lumping. Right: solution via Runge-Kutta scheme with mass lumping.
• Figure 5: Convergence rate of the central difference and Runge-Kutta scheme.

Theorems & Definitions (11)

• Remark 1
• Lemma 1
• Theorem 1
• proof
• Lemma 2
• Lemma 3
• Theorem 2
• Theorem 3
• proof
• proof