Optimized Schwarz waveform relaxation methods for wave-heat coupling in one dimensional bounded domains

We are interested in heterogeneous domain decomposition methods to couple partial differential equations in space-time. The coupling can be used to describe the exchange of heat or forces or both, and has important applications like fluid-structure or ocean-atmosphere coupling. Heterogeneous domain decomposition methods permit furthermore the reuse of existing codes which represent long term investments, a further great advantage in applications. We require that our method can use different and adaptive time steps for the different models, can be executed in parallel, is robust, and can use independent fast inner solvers. An ideal candidate is Optimized Schwarz Waveform Relaxation (OSWR) that can be used without overlap, which is important for the different physical models. We focus here on the model problem of coupling a heat and a wave equation in one spatial dimension, which we consider to be a minimal example of relevance, and our goal is to design and analyze transmission conditions such that OSWR converges as fast as possible. We propose two strategies, a first one where we optimize the transmission using one common parameter, and a second one where we use the wave characteristics of one subdomain to choose one parameter, and then optimize the other. We illustrate our results with numerical experiments.