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

2024/08/20 by Deeksha Tomer, Tomer, Deeksha, Basanti Mandal +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2408.11171

openalex publication_date 2024/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Algorithms for Parabolic Problems
• Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation for the Wave\n Equation
• Dirichlet-Neumann Waveform Relaxation Method for the 1D and 2D Heat and Wave Equations in Multiple subdomains
• Neumann-Neumann Waveform Relaxation Algorithm in Multiple subdomains for\n Hyperbolic Problems in 1D and 2D
• Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for time fractional sub-diffusion and diffusion-wave equations