Localized implicit time stepping for the wave equation

TL;DR

This paper introduces a localized implicit time stepping method for the wave equation that uses domain decomposition and localized subproblems, enabling parallel computation without inner iterations, and is supported by theoretical analysis and numerical tests.

Contribution

It proposes a novel localized implicit time stepping scheme for the wave equation that avoids inner iterations and supports parallel computation, backed by rigorous analysis.

Findings

Localized solutions closely approximate global implicit solutions

Method enables efficient parallel computation on overlapping subdomains

Numerical examples demonstrate the effectiveness of the approach

Abstract

This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a re-start is introduced after a certain amount of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented.