Optimized Schwartz Waveform Relaxation Method For Wave-heat Coupling Problem

As a simplified model of fluid-structure coupling problem,wave-heat coupling problem has been widely concerned in recent years.Chouly solved the wave-heat coupling problem in one-dimensional unbounded subdomains by using the optimized Schwarz method [N umerical Algorithms,90(2),2022,pp.631-668],and there was no time delay on the interface condition of the model.In practical application,however,many problems are affected by time delay,such as the wave-heat coupling problem with time delay on the interface studied by Peralta[Discrete & Continuous Dynamical Systems,38(6),2018,p.3055].It is worth noting that when the model has time delay on the interface,the solution of the system at the current time is influenced by the solution in the past time,so it is more complicated than the model without time delay.In this paper,we use the optimized Schwarz method to solve the wave-heat coupling problem with time delay on the interface.The optimized Schwarz method designs new transmission conditions on the interface by introducing relaxation parameters,then optimizes the relaxation parameters to speed up the convergence of the algorithm as much as possible.In this paper,we propose a Robin transmission condition and a two-sided Robin transmission condition.We prove that the optimized Schwarz method with Robin optimized transmission condition does not converge when solving the wave-heat coupling problem with time delay on the interface.Therefore,we focus on the discussion of optimized Schwarz method with two-sided Robin transmission condition and obtain the convergence factor by Fourier analysis.Unfortunately,due to the time delay on the interface,the monotonicity of the convergence factor about Fourier frequency and relaxation parameters is difficult to determine,then we transform the problem to solve the optimization problem of approximate convergence factor and obtain the convergence rate estimation under the optimal parameters.At this time,the optimized Schwarz method degenerates into the classical Schwarz method,so we propose to use the classical Schwarz method to solve the wave-heat coupling problem with time delay on the interface,and analyze the convergence of the algorithm.Finally,we illustrate the theoretical results of this paper through numerical experiments.