Error analysis of an implicit-explicit time discretization scheme for semilinear wave equations with application to multiscale problems

TL;DR

This paper introduces an unconditionally stable IMEX scheme for semilinear wave equations with damping, providing error bounds and demonstrating its effectiveness for multiscale problems with oscillating coefficients.

Contribution

The paper develops a novel IMEX scheme with proven error bounds and applies it to multiscale wave equations, showing convergence in complex heterogeneous settings.

Findings

Unconditionally stable IMEX scheme for semilinear wave equations.

Error bounds established for combined space and time discretization.

Demonstrated convergence for multiscale wave problems with oscillating coefficients.

Abstract

We present an implicit-explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly, we obtain a method which is not only unconditionally stable but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application, we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.