Non-overlapping Schwarz waveform relaxation for wave-heat interface coupling with time-delayed interaction

This paper investigates Schwarz waveform relaxation (SWR) methods for solving a coupled wave-heat system modelling time-delayed interaction between a structure and fluid, where the time delay significantly impacts stability properties. A non-overlapping SWR algorithm is proposed, utilizing a zeroth-order two-sided transmission condition. The free parameters in the transmission condition are optimized through Fourier analysis by examining a sharp upper bound of the corresponding convergence factor. The results indicate that the time-delayed interface interaction profoundly affects the algorithm's performance, causing the optimized transmission condition to revert to the original interface condition. Additionally, it is proven that, unlike in the case without time delay, the zeroth-order single-sided transmission condition leads to divergence of the SWR iteration. Numerical findings further reveal that local transmission operators are ineffective in accelerating the SWR iteration, particularly for large time delays. This highlights the need for further research into efficient transmission condition design.