Optimized Schwarz Waveform Relaxation for the Damped Wave Equation

TL;DR

The paper tackles slow Schwarz Waveform Relaxation convergence for the damped wave equation under viscoelastic damping, where classical absorbing interface conditions underperform. It introduces a two-parameter transmission operator with Λ = p ∂t + q and derives an explicit convergence factor ρ(s;p,q)=\frac{|κ(s)cos(κ(s)a)-(ps+q)sin(κ(s)a)|}{|κ(s)cos(κ(s)b)+(ps+q)sin(κ(s)b)|} with κ(s) = -i s/c sqrt((1+γ s^{-1})/(1+ν c^{-2} s)). Two optimization strategies, L_infty and L2, minimize ρ along the imaginary axis s=iω over [ω_min, ω_max], where ω_min=π/T and ω_max=π/Δt, using a discretized frequency grid and Nelder–Mead. Numerical experiments show that optimized parameters significantly speed up convergence compared with standard absorbing conditions, especially in the viscoelastic regime, and yield a computationally efficient alternative to exhaustive search. The framework thus offers robust acceleration for wave simulations in acoustics and related damped-wave problems.

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.

Figures (4)

• Figure 1: Performance surfaces showing the final SWR error (log-scale) after $k = 10$ iterations. Left column: increasing telegrapher damping $\gamma > 0$ with $\nu = 0$ (a$_\gamma$: $\gamma = 4$, b$_\gamma$: $\gamma = 10$). Right column: increasing viscoelastic damping $\nu > 0$ with $\gamma = 0$ (a$_\nu$: $\nu = 0.001$, b$_\nu$: $\nu = 0.05$).
• Figure 2: SWR error curve comparison when varying one damping coefficient at a time. The horizontal axis represents the iteration index k, and the vertical axis represents the relative SWR error on a logarithmic scale. First row: $L_\infty$ strategy. Second row: $L_2$ strategy. (a) Different values of $\gamma$ with $\nu = 0$. (b) Different values of $\nu$ with $\gamma = 0$. The dashed horizontal line indicates the discrepancy between the full-domain FDTD solution and the analytic ground truth.
• Figure 3: Predicted global convergence factors $\hat{\rho}_{\infty}$ (top row) and $\hat{\rho}_2$ (bottom row). (a) Contours of $\hat{\rho}$ in the $(p,q)$-plane. (b) Contours of $\hat{\rho}$ in the $(\gamma,\nu)$-plane (log–log axes).
• Figure 4: Comparison between the $L_{\infty}$-optimized (top row) and $L_2$-optimized (bottom row) transmission parameters. (a) Optimal $(p,q)$ points obtained by varying $\gamma$ or $\nu$. (b) Isolines of optimal $(p,q)$ for sweeps at fixed $\nu$. (c) Isolines of optimal $(p,q)$ for sweeps at fixed $\gamma$.