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Neumann-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay
Deeksha Tomer
TL;DR
Parallel numerical methods for hyperbolic PDEs with time delay remain understudied, despite the challenge of solving these systems on large or heterogeneous domains. This paper applies NNWR to delayed wave equations on nonuniform subdomains, establishing a linear convergence estimate and finite-step convergence under a time-window condition. Numerical experiments support the theoretical results.
Problem
Parallel numerical methods for hyperbolic PDEs with time delays remain largely unexplored, while solving such systems is non-trivial for large or heterogeneous spatial domains.
Source: §1 · p. 1Method
The paper applies NNWR with Neumann interface transmission to delayed wave equations on nonuniform subdomains, analyzing convergence with Fourier and Laplace transform techniques.
Source: §1 · p. 1, Step II: Solve Neumann subproblems on… · pp. 2–3Results
For θ = 1/4, NNWR has a linear convergence estimate and converges in at most k + 1 iterations when T/k ≤ 4 min {a/c, b/c} for asymmetric two-subdomain decompositions.
Source: §2.1 · p. 4, §2.2 · p. 6Takeaways & Limitations
The analyses and numerical experiments support NNWR convergence for delayed wave equations on asymmetric subdomains, with delay terms adding time lag without affecting convergence.
Source: §2.2 · p. 8, §4 · p. 10The analysis is scoped to a linear second-order wave equation with constant delay τ.
Source: Step II: Solve Neumann subproblems on… · p. 2
Abstract
from arXiv · showhide
Hyperbolic PDEs with time delay are essential for modeling a wide range of real-world applications. Since closed-form solutions are often not attainable, developing accurate and efficient numerical methods becomes crucial. In this work a novel substructuring waveform relaxation method namely Neumann-Neumann Waveform Relaxation (NNWR) has been implemented for numerically solving hyperbolic delay PDEs in the case of non-uniform subdomain partitioning. The convergence behavior is first analyzed using Fourier analysis to obtain an estimate, followed by a demonstration of finite-step convergence through Laplace transform techniques. Various test cases are presented as numerical illustrations.
1 Introduction
Parallel methods for hyperbolic PDEs with time delay remain understudied, motivating domain decomposition for large or heterogeneous spatial domains. The paper applies NNWR to delayed wave equations on nonuniform subdomains and outlines its iterative procedure and convergence analysis.
1 Introduction
- Parallel numerical methods for time-delayed hyperbolic PDEs remain largely unexplored, especially for large or heterogeneous spatial domains.The passage motivates domain decomposition as a route to parallel computation and scalability.Source: §1 · p. 1
- NNWR uses Neumann transmission at subdomain interfaces and is presented as suitable for hyperbolic problems and nonuniform multiple subdomains.Waveform relaxation is described as highly amenable to parallelization.Source: §1 · p. 1
NNWR Iterative Procedure
- The iterative procedure solves independent Dirichlet subproblems in each subdomain before applying a Neumann-based interface correction.The procedure is introduced for the classical wave equation on a domain split into subdomains.Source: §1 · pp. 1–2
- The model is a linear second-order wave equation with constant delay τ, and θ∈(0,1] is the relaxation parameter for updates from initial interface guess l0.The equation includes the delayed term λv(x, t − τ); c denotes wave speed and λ is freely chosen.Source: Step II: Solve Neumann subproblems on… · pp. 2–3
- The work estimates convergence with Fourier analysis, derives finite-step convergence conditions with Laplace transforms, and evaluates the method numerically.Well-posedness is addressed in prior studies.Source: Step II: Solve Neumann subproblems on… · p. 3
2 Analysis of NNWR for Asymmetric Decomposition
For asymmetric two-subdomain decompositions, NNWR has a Fourier-based linear convergence estimate and a Laplace-based finite-step convergence condition. The finite-step result holds for a bounded time window and is reported to remain analogous to the no-delay wave PDE case.
2 Analysis of NNWR for Asymmetric Decomposition
- For unequal two-subdomain splits, NNWR solves subdomain Dirichlet error problems, follows with Neumann subproblems, and updates across the interface.The interface guess is q0(t).Source: §2 · p. 3
2.1 Fourier-Based Convergence Estimate
- With θ = 1/4, the Fourier analysis establishes linear convergence for the time-delayed hyperbolic PDE on t ∈ (0, ∞).The analysis concerns asymmetric subdomains.Source: §2 · p. 3, §2.1 · p. 4
2.2 Laplace Transform Based Convergence
- At most k + 1 iterations suffice when T/k ≤ 4 min {a/c, b/c}, with θ = 1/4 for the constant-delay PDE.Here a and b are subdomain sizes, and c is wave speed.Source: §2.2 · pp. 6, 8
3 Numerical Illustration
The numerical illustrations compare NNWR convergence across nonuniform decompositions and time-window choices. In the reported asymmetric 2D case, the optimal relaxation parameter is θ = 1/4.
3 Numerical Illustration
- The numerical discretization uses a Leapfrog scheme with ∆t = 2 × 10−2 and ∆x = 0.1 on uniformly spaced grid points.The computational domain is divided into nonuniform subdomains.Source: §3 · p. 8
- Figure 1 compares numerical NNWR convergence with theoretical bounds for two nonuniform subdomain splits at θ = 1/4.The splits are D1 = (0, 4), D2 = (4, 6) and D1 = (0, 2.5), D2 = (2.5, 6).Source: Figure 1 · p. 8, §3 · p. 9
- Figure 2 presents NNWR convergence trends for different time-window lengths T at θ = 1/4.The interface initial value is q0(t) = t2 for t ∈ (0, T].Source: Figure 2 · p. 9, §3 · p. 10
- Figure 3 illustrates the inverse Laplace transform yielding F(t) = cos(t2) and its time-shifted version.The shifted transform is L−1{e−s F̂(s)}.Source: Figure 3 · p. 9
- For NNWR’s asymmetric 2D split, D1 = (0, 6) × (0, 8) and D2 = (6, 8) × (0, 8), the optimal relaxation parameter is θ = 1.Figure 4 shows convergence results for the 2D hyperbolic PDE with time delay at different θ values.Source: Figure 4 · p. 9, §3 · p. 10
4 Conclusion
The conclusion reports convergence estimates for NNWR on hyperbolic PDEs with time delay in two asymmetric subdomains. Fourier analysis gives a linear convergence bound, while Laplace analysis establishes finite-step convergence at θ = 1/4, supported by numerical experiments.
4 Conclusion
- For hyperbolic PDEs with time delay on two asymmetric subdomains, Fourier analysis establishes a linear convergence bound and Laplace analysis demonstrates finite-step convergence at θ = 1/4.The theoretical results are supported by numerical experiments.Source: §4 · p. 10