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

Deeksha Tomer, Bankim Chandra Mandal

TL;DR

This work extends Dirichlet-Neumann Waveform Relaxation (DNWR) and Neumann-Neumann Waveform Relaxation (NNWR) to PDEs with time delay, analyzing parabolic, hyperbolic, and neutral models. By applying Laplace-transform techniques to the error equations, it derives explicit convergence factors: for symmetric two-subdomain partitions DNWR converges in two iterations at $\theta=\tfrac{1}{2}$ and NNWR at $\theta=\tfrac{1}{4}$, with linear convergence for other $\theta$ values; analogous results hold for hyperbolic and neutral PDEs, with corresponding expressions $\hat{h}^k(s)=(1-2\theta)^k\hat{h}^0(s)$ or $\hat{g}^k(s)=(1-4\theta)^k\hat{g}^0(s)$. Numerical illustrations for reaction-diffusion, wave, and neutral PDEs corroborate the theory and demonstrate effective parallelization, including multi-subdomain scenarios, and show superiority over classical/optimized Schwarz methods for optimal $\theta$ choices. The findings establish DNWR and NNWR as efficient, scalable tools for time-delayed PDEs, offering rapid convergence and parallel performance across a range of models. This contributes a practical framework for solving Delay PDEs in large-scale computations where interface-driven iteration is advantageous.

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.

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

TL;DR

This work extends Dirichlet-Neumann Waveform Relaxation (DNWR) and Neumann-Neumann Waveform Relaxation (NNWR) to PDEs with time delay, analyzing parabolic, hyperbolic, and neutral models. By applying Laplace-transform techniques to the error equations, it derives explicit convergence factors: for symmetric two-subdomain partitions DNWR converges in two iterations at and NNWR at , with linear convergence for other values; analogous results hold for hyperbolic and neutral PDEs, with corresponding expressions or . Numerical illustrations for reaction-diffusion, wave, and neutral PDEs corroborate the theory and demonstrate effective parallelization, including multi-subdomain scenarios, and show superiority over classical/optimized Schwarz methods for optimal choices. The findings establish DNWR and NNWR as efficient, scalable tools for time-delayed PDEs, offering rapid convergence and parallel performance across a range of models. This contributes a practical framework for solving Delay PDEs in large-scale computations where interface-driven iteration is advantageous.

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.
Paper Structure (16 sections, 6 theorems, 64 equations, 12 figures)

This paper contains 16 sections, 6 theorems, 64 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Parabolic PDE with Time Delay
  3. Convergence of DNWR
  4. Convergence of NNWR
  5. Hyperbolic PDEs with Time Delay
  6. Convergence Results of DNWR :
  7. Convergence results of NNWR
  8. Neutral PDE
  9. DNWR for Neutral PDE
  10. NNWR for Neutral PDE
  11. Numerical Illustrations
  12. Reaction-Diffusion Equation with Time Delay
  13. Wave equation with Time Delay
  14. Neutral PDE
  15. Experiments for Multiple Subdomain case
  16. ...and 1 more sections

Key Result

Theorem 1

For the Parabolic PDE (eq_1) with time delay, the DNWR algorithm (eq_2)-(eq_33) converges linearly for $0<\theta<1$, $\theta\neq 1/2$. For $\theta=1/2$ it converges within two iterations, irrespective of the time window.

Figures (12)

  • Figure 1: Parabolic PDE with time delay when $a_1 \neq 0$: Convergence for different relaxation parameters; Left: DNWR algorithm, Right: NNWR algorithm.
  • Figure 2: Parabolic PDE with time delay when $a_1 \neq 0$: Left; Convergence of DNWR for different parameters for larger Dirichlet domain; Right: Convergence of NNWR for different parameters.
  • Figure 3: Parabolic PDE with time delay when $a_1\neq0$; Comparison of NNWR and DNWR with classical Schwarz and Optimised Schwarz method.
  • Figure 4: Parabolic PDE with time delay when $a_1=0$; Left: Convergence of DNWR for different $\theta$; Right: Convergence of NNWR for different $\theta$.
  • Figure 5: Parabolic PDE with time delay when $a_1=0$; Left: Convergence of DNWR for larger Dirichlet domain, Right: Convergence of NNWR.
  • ...and 7 more figures

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 2 more