Domain splitting with localized implicit prediction for wave equations
Abstract
We propose and analyze a non-iterative domain decomposition time integrator for the linear acoustic wave equation. The method is based on an overlapping decomposition in space and uses a localized implicit prediction to provide artificial boundary data at the interfaces. We use conforming finite elements of arbitrary order to discretize in space and allow spatially varying material coefficients. We prove an error bound relative to the global Crank--Nicolson approximation that decays exponentially with the size of both the subdomain and prediction overlaps. Compared to earlier work, this new method avoids a CFL-type step size restriction and allows the use of finite elements of arbitrary order. The analysis combines a discrete Saint-Venant principle with new decay and truncation estimates for the localized prediction. Numerical experiments support the analysis and demonstrate stable behavior for all tested time step sizes, accuracy comparable to that of the global Crank--Nicolson approximation, and the potential for efficient parallel execution.
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BibTeX
@misc{buchholz2026domain,
title = {{Domain splitting with localized implicit prediction for wave equations}},
author = {Tim Buchholz and Roland Maier},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13822},
eprint = {2609.13822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13822},
}
RIS
TY - UNPB
TI - Domain splitting with localized implicit prediction for wave equations
AU - Tim Buchholz
AU - Roland Maier
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13822
UR - https://arxiv.org/abs/2609.13822
AB - We propose and analyze a non-iterative domain decomposition time integrator for the linear acoustic wave equation. The method is based on an overlapping decomposition in space and uses a localized implicit prediction to provide artificial boundary data at the interfaces. We use conforming finite elements of arbitrary order to discretize in space and allow spatially varying material coefficients. We prove an error bound relative to the global Crank--Nicolson approximation that decays exponentially with the size of both the subdomain and prediction overlaps. Compared to earlier work, this new method avoids a CFL-type step size restriction and allows the use of finite elements of arbitrary order. The analysis combines a discrete Saint-Venant principle with new decay and truncation estimates for the localized prediction. Numerical experiments support the analysis and demonstrate stable behavior for all tested time step sizes, accuracy comparable to that of the global Crank--Nicolson approximation, and the potential for efficient parallel execution.
ER -