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Dirichlet-Neumann Waveform Relaxation Method with Multiple Subdomains for Reaction-Diffusion Equation with a Time Delay

Bankim C. Mandal, Deeksha Tomer

TL;DR

The paper tackles solving a delayed reaction-diffusion PDE on a domain decomposed into multiple subdomains using Dirichlet-Neumann Waveform Relaxation (DNWR). It extends DNWR to multi-subdomain settings and evaluates three transmission arrangements through a 1-D, five-subdomain test case, emphasizing interface data exchange and convergence. A key finding is that arrangement 3 delivers the best efficiency for longer time domains, with a relaxation parameter of θ = 1/2 generally yielding the fastest convergence, while the iteration count increases with the number of subdomains. The work demonstrates a scalable, partially parallelizable DNWR strategy for delayed reaction-diffusion problems and offers guidance on interface design to enhance convergence in domain-decomposed solvers.

Abstract

In this study, we present the numerical investigation of the Dirichlet-Neumann Waveform Relaxation (DNWR) algorithm applied to multiple subdomains for the reaction-diffusion equation with time delay. Various arrangements of transmission conditions between subdomains are explored and a series of numerical experiments are conducted to evaluate and compare the efficiency and effectiveness of these configurations.

Dirichlet-Neumann Waveform Relaxation Method with Multiple Subdomains for Reaction-Diffusion Equation with a Time Delay

TL;DR

The paper tackles solving a delayed reaction-diffusion PDE on a domain decomposed into multiple subdomains using Dirichlet-Neumann Waveform Relaxation (DNWR). It extends DNWR to multi-subdomain settings and evaluates three transmission arrangements through a 1-D, five-subdomain test case, emphasizing interface data exchange and convergence. A key finding is that arrangement 3 delivers the best efficiency for longer time domains, with a relaxation parameter of θ = 1/2 generally yielding the fastest convergence, while the iteration count increases with the number of subdomains. The work demonstrates a scalable, partially parallelizable DNWR strategy for delayed reaction-diffusion problems and offers guidance on interface design to enhance convergence in domain-decomposed solvers.

Abstract

In this study, we present the numerical investigation of the Dirichlet-Neumann Waveform Relaxation (DNWR) algorithm applied to multiple subdomains for the reaction-diffusion equation with time delay. Various arrangements of transmission conditions between subdomains are explored and a series of numerical experiments are conducted to evaluate and compare the efficiency and effectiveness of these configurations.
Paper Structure (7 sections, 13 equations, 10 figures)

This paper contains 7 sections, 13 equations, 10 figures.

Table of Contents

  1. Introduction
  2. DNWR for Multiple Subdomains - Numerical Experiments
  3. Arrangement 1
  4. Arrangement $2$
  5. Arrangement $3$
  6. Additional Numerical Experiments
  7. Conclusion

Figures (10)

  • Figure 1: Arrangement 1 of Boundary conditions for DNWR in multisubdomain setup
  • Figure 2: Convergence result of DNWR algorithm for arrangement $1$. Left: DNWR for shorter time window; Right: DNWR for longer time window
  • Figure 3: Arrangement 2 of boundary conditions for DNWR in multisubdomain setup
  • Figure 4: Convergence result of DNWR algorithm for arrangement $2$. Left: DNWR for shorter time window; Right: DNWR for longer time window.
  • Figure 5: Arrangement 3 of boundary conditions in multisubdomain setup
  • ...and 5 more figures