A First Closed-Form Robin Parameter for Optimized Schwarz Waveform Relaxation as a Function of Iteration Count, Overlap, and Time Window Length
Abstract
Schwarz waveform relaxation (SWR) is a parallel-in-time iterative method for the solution of evolution problems and belongs to the class of Parallel-in-Time algorithms that have received increasing attention over the past three decades. For parabolic problems, SWR converges linearly over long time intervals and exhibits superlinear convergence on short time intervals. Since the introduction of optimized variants (OSWR) nearly three decades ago, determining the optimized Robin parameter when OSWR is applied on bounded time windows has remained an open problem. We introduce here a new analytical approach that, for the first time, yields a closed-form expression for the optimal Robin parameter in the transmission conditions for the one-dimensional heat equation with two overlapping subdomains. The resulting formula explicitly characterizes the dependence of the optimal parameter on the iteration number, the overlap size, and the length of the time window. Beyond resolving this long-standing optimization problem, our analysis provides a framework that can be extended to more general settings, including higher-dimensional problems, multiple subdomains, and higher-order transmission conditions.
Related
- Extending a new two-grid waveform relaxation on a spatial finite element discretization
- Neumann-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay
- Dirichlet-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay in Multiple Subdomains
- Dirichlet--Neumann waveform relaxation for heterogeneous heat equations: fully discrete l2 analysis
- Dirichlet-Neumann waveform relaxation for heterogeneous heat equations: continuous and time discrete L2 analysis
BibTeX
@misc{rodriguez2026first,
title = {{A First Closed-Form Robin Parameter for Optimized Schwarz Waveform Relaxation as a Function of Iteration Count, Overlap, and Time Window Length}},
author = {Alejandro Alfonso Rodriguez and Martin J. Gander and Felix Kwok},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05576},
eprint = {2611.05576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05576},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A First Closed-Form Robin Parameter for Optimized Schwarz Waveform Relaxation as a Function of Iteration Count, Overlap, and Time Window Length
AU - Alejandro Alfonso Rodriguez
AU - Martin J. Gander
AU - Felix Kwok
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05576
UR - https://arxiv.org/abs/2611.05576
AB - Schwarz waveform relaxation (SWR) is a parallel-in-time iterative method for the solution of evolution problems and belongs to the class of Parallel-in-Time algorithms that have received increasing attention over the past three decades. For parabolic problems, SWR converges linearly over long time intervals and exhibits superlinear convergence on short time intervals. Since the introduction of optimized variants (OSWR) nearly three decades ago, determining the optimized Robin parameter when OSWR is applied on bounded time windows has remained an open problem. We introduce here a new analytical approach that, for the first time, yields a closed-form expression for the optimal Robin parameter in the transmission conditions for the one-dimensional heat equation with two overlapping subdomains. The resulting formula explicitly characterizes the dependence of the optimal parameter on the iteration number, the overlap size, and the length of the time window. Beyond resolving this long-standing optimization problem, our analysis provides a framework that can be extended to more general settings, including higher-dimensional problems, multiple subdomains, and higher-order transmission conditions.
ER -