El-WaveHoltz: A Time-Domain Iterative Solver for Time-Harmonic Elastic Waves

TL;DR

This paper introduces El-WaveHoltz, an iterative time-domain solver for elastic wave equations that improves accuracy by removing time discretization errors and demonstrates effective convergence properties.

Contribution

It develops a novel implicit time-stepping scheme for the WaveHoltz iteration, enhancing accuracy and practical performance for elastic wave problems.

Findings

The implicit scheme removes time discretization error.

Convergence rate depends on the shortest shear wave speed.

Implicit scheme benefits meshes with varying element sizes.

Abstract

We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that filters the solution of the wave equation with time-harmonic forcing and boundary data. As in the original WaveHoltz method, we reformulate the fixed point iteration as a positive definite linear system of equations that is iteratively solved by a Krylov method. We present two time-stepping schemes, one explicit and one (novel) implicit, which completely remove time discretization error from the WaveHoltz solution by performing a simple modification of the initial data and time-stepping scheme. Numerical experiments indicate an iteration scaling similar to that of the original WaveHoltz method, and that the convergence rate is dictated by the shortest (shear) wave speed of the problem. We additionally show that the implicit scheme can be advantageous in practice for meshes with disparate element sizes.