Dirichlet–Neumann and Neumann–Neumann waveform relaxation methods for PDEs with time delay

Abstract We introduce and compare two domain decomposition-based numerical methods, namely the Dirichlet–Neumann and Neumann–Neumann waveform relaxation methods (DNWR and NNWR, respectively), tailored for solving partial differential equations (PDEs) incorporating time delay. Time delay phenomena frequently arise in various real-world systems, making their accurate modelling and simulation crucial for understanding and prediction. We consider a series of model problems, ranging from Parabolic, Hyperbolic to Neutral PDEs with time delay and apply the iterative techniques DNWR and NNWR for solving in parallel. We present the theoretical foundation, numerical implementation and comparative performance analysis of these two methods. Through numerical experiments and simulations, we explore their convergence properties, computational efficiency and applicability to various types of PDEs with time delay.