Dirichlet-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay in Multiple Subdomains

1 Introduction

• Parallelizing delayed hyperbolic PDEs remains a significant challenge for large, complex numerical systems.Such equations model systems involving wave propagation, vibrations, and oscillations.Source: §1 · p. 2
• DNWR is a non-overlapping domain-decomposition method that updates interface conditions iteratively using Dirichlet and Neumann solves.The method can use different numerical schemes in different subdomains and can act as a direct solver for suitably chosen time windows.Source: §1 · p. 2
• The study extends DNWR from two subdomains to a multisubdomain framework for hyperbolic delay PDEs.The model problem is a linear wave equation with constant time delay.Source: §1 · p. 2
• The work investigates DNWR convergence for multisubdomain problems in both one and two spatial dimensions.It also develops a rigorous convergence analysis beyond one-dimensional configurations.Source: §1 · p. 3
• The paper compares DNWR performance with NNWR, OSWR, and Classical SWR.The introduction identifies these as alternative domain-decomposition frameworks for the comparative analysis.Source: §1 · p. 3

2 DNWR Formulation for multiple subdomain case

• Arrangement 3 is selected for convergence study after outperforming Arrangements 1 and 2 in cited reaction-diffusion and heat-equation studies.The reaction-diffusion comparison reports greater iteration efficiency, and the classical heat-equation comparison reports better convergence.Source: §2 · p. 3
• The domain is divided into N subdomains, with the middle subdomain solved first and adjacent Dirichlet–Neumann subproblems handled sequentially outward.The middle subdomain is indexed by m = ⌈N/2⌉.Source: §2 · pp. 3–4
• Interface updates use a relaxation parameter θ in the interval (0, 1].The formulation applies iteratively across the subdomains.Source: §2 · pp. 3–4

• The convergence analysis uses θ = 1/2, identified as the optimal relaxation parameter from prior studies.The cited studies are.Source: §3 · p. 5
• At most k + 1 iterations suffice when T/k ≤ dmin/c, where dmin is the minimal subdomain width and c is the wave speed.The convergence analysis assumes θ = 1/2.Source: §3 · p. 5
• The proof applies a Laplace transform to the error equations and expresses interface-error coefficients through exponential expansions and time shifts.The analysis uses geometric-series expansions and the inverse-Laplace shifting property to show the errors vanish.Source: §3 · pp. 5, 9, 10

• The spatial domain is split into vertical strips, and a Fourier sine series in y decouples the two-dimensional problem into independent one-dimensional problems for each mode.The strips have widths di, with dmin defined as their minimum width.Source: §4 · pp. 10–12
• The 2D DNWR iteration is represented as a combination of cuboids, corresponding to evolving spatial strips in the spacetime decomposition.Fig. 3 depicts the evolution of 2D spatial strips over time as a decomposition of the total spacetime volume.Source: §4 · p. 11, Figure 3 · p. 11
• The analysis retains only the leading term of an exponential expansion because the λ-term introduces an additional delay effect.Higher-order contributions involving λ are neglected in this step.Source: §4 · p. 13
• Whenever t < kdmin/c, one additional iteration eliminates all interface discrepancies and recovers the exact solution across the spatial domain.The theorem states convergence in at most k + 1 iterations for fixed T when its stated condition holds.Source: §4 · pp. 12–13

5.1 DNWR in 1D Mutisubdomain

• For a 1D run with T = 3 and dmin = 0.5, the convergence bound is at most 7 iterations.The experiments use a Leapfrog scheme and choose the delay as an integer multiple of the time step.Source: §5 · p. 13, §5.1 · p. 14
• For 3, 5, 7, and 9 subdomains at T = 12 and dmin = 1, the maximum iteration count is at most k + 1, matching Theorem 1.Source: §5.1 · p. 14
• For multi-subdomain one- and two-dimensional problems, the convergence condition is T/k < dmin/c and is independent of delay term τ, so delay magnitude does not influence convergence behavior.Numerical experiments report exactly the same convergence results for Arrangements 1 and 2; Arrangement 1 is sequential, while Arrangements 2 and 3 are partially parallel.Source: §5.1 · pp. 14, 16

5.2 DNWR in 2D Multisubdomain

• The 2D experiments use three subdomains and compare minimum lengths dmin = 0.5 and dmin = 1.The corresponding convergence plots are shown for Arrangement 3.Source: §5.2 · p. 16, Figure 10 · p. 17

5.3 Comparative Study

• In three-subdomain tests, DNWR and NNWR converge faster than classical SWR in both 1D and 2D, and both outperform optimized SWR in 2D.Classical SWR converges only with overlap; in 1D, optimized SWR has a convergence rate comparable to DNWR and NNWR.Source: §5.3 · p. 17

6 Conclusion

• Laplace-transform analysis establishes finite-step convergence of multisubdomain DNWR for hyperbolic PDEs with time delays.Source: §6 · pp. 17–18
• The comparison covers DNWR, NNWR, classical SWR, and optimized SWR.Source: Figure 11 · p. 18, Table 1 · p. 18
• DNWR converges in fewer iterations than classical SWR, while NNWR converges better than DNWR for larger time intervals but requires roughly twice the work.NNWR's extra work comes from additional Dirichlet and correction steps.Source: §6 · p. 18

Data availability

• All data produced or generated during the research are included in the manuscript.Source: Data availability · p. 19