Optimized Red–Black Waveform Relaxation for the Damped Wave Equation
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We present an optimized Red–BlackWaveform Relaxation (WR) method for the one-dimensional damped wave equation with combined telegrapher damping (∂tu) and viscoelastic damping (∂t∂xxu). We leverage domain decomposition theory for hyperbolic–parabolic systems through three key components: (1) parallel decomposition across N overlapping subdomains, (2) red–black partitioning to alternate subdomain updates, and (3) optimized Robin transmission conditions at each interface. We combine frequency-domain analysis with time-aware error bounds derived from Green’s-function kernels, explicitly investigating how the damping parameters influence convergence. We demonstrate through numerical experiments that our optimized interface conditions reduce iterative error substantially faster than classical Dirichlet coupling and confirm the method’s efficacy for wave-dominated systems with mixed dissipation mechanisms.