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 modeling 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 foundations, 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.

翻译：本文介绍并比较了两种基于区域分解的数值方法——Dirichlet-Neumann波形松弛方法（DNWR）与Neumann-Neumann波形松弛方法（NNWR），这两种方法专为求解含时滞的偏微分方程（PDEs）而设计。时滞现象广泛存在于各类实际系统中，因此对其进行精确建模与仿真对于系统理解和预测至关重要。我们考虑了一系列模型问题，涵盖抛物型、双曲型及含时滞的中性偏微分方程，并应用迭代技术DNWR与NNWR进行并行求解。本文阐述了这两种方法的理论基础、数值实现及性能对比分析。通过数值实验与仿真，我们深入探究了它们的收敛特性、计算效率以及对各类含时滞偏微分方程的适用性。