[Paper Review] Dirichlet-Neumann Waveform Relaxation Method for the 1D and 2D Heat and Wave Equations in Multiple subdomains

Martin J. Gander, Félix Kwok+1 more|arXiv (Cornell University)|Jul 14, 2015

Advanced Numerical Methods in Computational Mathematics14 references3 citations

TL;DR

This paper introduces a Dirichlet-Neumann Waveform Relaxation (DNWR) method for solving 1D and 2D heat and wave equations via non-overlapping domain decomposition in space-time, using iterative subdomain solves with alternating Dirichlet and Neumann interface conditions. For the heat equation, superlinear convergence is proven with a relaxation parameter θ=1/2 over finite time windows; for the wave equation, finite step convergence is achieved under similar conditions, with faster convergence than optimized Schwarz WR methods.

ABSTRACT

We present a Waveform Relaxation (WR) version of the Dirichlet-Neumann algorithm, formulated specially for multiple subdomains splitting for general parabolic and hyperbolic problems. This method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves in space-time with corresponding interface condition, and finally organize an exchange of information between neighboring subdomains. Using a Fourier-Laplace transform argument, for a particular relaxation parameter, we present convergence analysis of the algorithm for the heat and wave equations. We prove superlinear convergence for finite time window in case of the heat equation, and finite step convergence for the wave equation. The convergence behavior however depends on the size of the subdomains and the time window length on which the algorithm is employed. We illustrate the performance of the algorithm with numerical results, and show a comparison with classical and optimized Schwarz WR methods.

Motivation & Objective

• To extend the Dirichlet-Neumann Waveform Relaxation (DNWR) method from two to multiple subdomains for parabolic and hyperbolic PDEs.
• To analyze convergence behavior of DNWR for the 1D heat and wave equations using Fourier-Laplace transforms.
• To generalize the DNWR framework to 2D wave equations and validate convergence properties numerically.
• To compare DNWR performance with optimized Schwarz Waveform Relaxation (SWR) and Neumann-Neumann WR (NNWR) in terms of convergence speed and computational cost.
• To demonstrate robustness of DNWR under non-uniform time grids and discontinuous wave speeds across subdomain interfaces.

Proposed method

• Formulates a non-overlapping space-time domain decomposition method using alternating Dirichlet and Neumann conditions at subdomain interfaces.
• Solves subdomain problems iteratively: first with Dirichlet conditions on interfaces, then with Neumann conditions, using waveform relaxation over a finite time window.
• Applies a relaxation parameter θ ∈ (0,1] to stabilize and accelerate convergence, with θ=1/2 shown to yield optimal convergence for both heat and wave equations.
• Employs Fourier-Laplace transform analysis to derive convergence rates for the 1D heat and wave equations.
• Uses numerical implementations with finite difference spatial discretization and non-uniform time stepping to validate theoretical results.
• Applies time projection techniques for inter-subdomain data exchange when using non-uniform time grids, ensuring consistency with linear complexity via merge-sort-like algorithms.

Experimental results

Research questions

• RQ1 What is the convergence behavior of the DNWR method for the 1D heat equation with multiple subdomains and a finite time window?
• RQ2 Does the DNWR method achieve finite step convergence for the 1D wave equation, and under what conditions on the time window and subdomain size?
• RQ3 How does the DNWR method extend to two-dimensional wave equations, and what convergence properties are observed?
• RQ4 How does DNWR compare in convergence speed and computational cost to optimized Schwarz Waveform Relaxation (SWR) and Neumann-Neumann WR (NNWR)?
• RQ5 Can DNWR maintain convergence and stability when subdomains use non-uniform time grids and discontinuous wave speeds?

Key findings

• For the 1D heat equation, superlinear convergence is proven with a relaxation parameter θ=1/2 over a finite time window T, under the condition T ≤ kh_min/c.
• For the 1D wave equation, finite step convergence is achieved in at most two iterations when T ≤ kh_min/c, with θ=1/2.
• In 2D wave equations, the DNWR method achieves finite step convergence under the condition T < kh_min/c, with similar behavior to the 1D case.
• Numerical results show that DNWR converges faster than optimized Schwarz Waveform Relaxation (SWR), especially in higher dimensions.
• While NNWR converges faster than DNWR, it requires nearly double the computational cost per iteration due to solving both Dirichlet and Neumann subproblems per subdomain.
• The DNWR method remains effective under non-uniform time grids and discontinuous wave speeds, with convergence demonstrated in three-step iterations for T=2.