Optimized Schwarz Waveform Relaxation for the Damped Wave Equation

TL;DR

This paper develops an optimized transmission operator for Schwarz Waveform Relaxation to improve convergence speed in damped wave equations, especially with viscoelastic damping, through frequency analysis and parameter optimization.

Contribution

Introduces a two-parameter transmission operator and optimization strategies to enhance Schwarz Waveform Relaxation for damped wave equations.

Findings

Optimized parameters significantly improve convergence speed.

Method outperforms standard absorbing conditions in viscoelastic damping cases.

Provides an efficient alternative to exhaustive parameter search.

Abstract

The performance of Schwarz Waveform Relaxation is critically dependent on the choice of transmission conditions. While classical absorbing conditions work well for wave propagation, they prove insufficient for damped wave equations, particularly in viscoelastic damping regimes where convergence becomes prohibitively slow. This paper addresses this limitation by introducing a more general transmission operator with two free parameters for the one-dimensional damped wave equation. Through frequency-domain analysis, we derive an explicit expression for the convergence factor governing the convergence rate. We propose and compare two optimization strategies (L-infinity and L-2 minimization) for determining optimal transmission parameters. Numerical experiments demonstrate that our optimized approach significantly accelerates convergence compared to standard absorbing conditions, especially for viscoelastic damping cases. The method provides a computationally efficient alternative to exhaustive parameter search while maintaining robust performance across different damping regimes.