Dirichlet--Neumann waveform relaxation for heterogeneous heat equations: fully discrete l2 analysis
Abstract
We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet--Neumann waveform relaxation. This allows the subproblems to be solved using separate codes, a so called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. Here, we use an exponentially weighted Fourier technique to derive new error estimates in l2 for finite time T in the fully discrete setting. These describe both linear and superlinear behavior. We show that the fully discrete estimate is close to a previously obtained time discrete estimate and independent of Δx1 and Δx2 when the CFL number in the subsolvers is large. We also show that the convergence behaviour depends on the ratio Δx1/Δx2 when the CFL number is small. Our numerical experiments show that the fully discrete estimate is accurate across a wide range of grid sizes Δx1, Δx2, time step sizes Δt and T.
Related
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BibTeX
@misc{birken2026dirichletneumann,
title = {{Dirichlet--Neumann waveform relaxation for heterogeneous heat equations: fully discrete $l^2$ analysis}},
author = {Philipp Birken and Niklas Kotarsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01475},
eprint = {2610.01475},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01475},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dirichlet--Neumann waveform relaxation for heterogeneous heat equations: fully discrete l^2 analysis
AU - Philipp Birken
AU - Niklas Kotarsky
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01475
UR - https://arxiv.org/abs/2610.01475
AB - We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet--Neumann waveform relaxation. This allows the subproblems to be solved using separate codes, a so called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. Here, we use an exponentially weighted Fourier technique to derive new error estimates in l^2 for finite time T in the fully discrete setting. These describe both linear and superlinear behavior. We show that the fully discrete estimate is close to a previously obtained time discrete estimate and independent of Δx_1 and Δx_2 when the CFL number in the subsolvers is large. We also show that the convergence behaviour depends on the ratio Δx_1/Δx_2 when the CFL number is small. Our numerical experiments show that the fully discrete estimate is accurate across a wide range of grid sizes Δx_1, Δx_2, time step sizes Δt and T.
ER -