Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Methods for PDEs with Time Delay

TL;DR

This paper introduces and compares Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation methods for efficiently solving PDEs with time delay, demonstrating their convergence and computational advantages across different PDE types.

Contribution

The paper presents novel domain decomposition methods tailored for PDEs with time delay, including theoretical analysis and practical comparison of DNWR and NNWR techniques.

Findings

Both methods show effective convergence for PDEs with time delay.

Numerical experiments highlight differences in computational efficiency.

Methods are applicable to parabolic, hyperbolic, and neutral PDEs with 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 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.