Dirichlet-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay in Multiple Subdomains
Abstract
Hyperbolic partial differential equations (PDEs) with time delay are essential mathematical tools used to model numerous physical systems where wave propagation or oscillations depend heavily on their historical states. As these applications scale in physical complexity, efficient parallel computing techniques become essential; however, developing highly scalable parallel solvers for delayed PDEs remains a significant computational challenge. To address this gap, this study extends the Dirichlet-Neumann Waveform Relaxation (DNWR) method to a multi-subdomain framework explicitly designed for solving time-delayed hyperbolic PDEs in parallel. We advance the underlying mathematical framework and provide a rigorous convergence analysis for both one-dimensional and two-dimensional spatial configurations. A central theoretical contribution of this work is the establishment of finite-step convergence for the multi-domain DNWR algorithm. These theoretical guarantees are firmly corroborated through comprehensive numerical experiments. Furthermore, a detailed comparative analysis against alternative domain decomposition techniques namely NNWR, OSWR, and Classical SWR, highlight the advantages of the proposed method. Finally, this approach provides a robust, highly parallelizable solver capable of efficiently handling complex hyperbolic PDEs with time delay.
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BibTeX
@misc{mandal2026dirichletneumann,
title = {{Dirichlet-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay in Multiple Subdomains}},
author = {Bankim Chandra Mandal and Deeksha Tomer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11120},
eprint = {2609.11120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11120},
}
RIS
TY - UNPB
TI - Dirichlet-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay in Multiple Subdomains
AU - Bankim Chandra Mandal
AU - Deeksha Tomer
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11120
UR - https://arxiv.org/abs/2609.11120
AB - Hyperbolic partial differential equations (PDEs) with time delay are essential mathematical tools used to model numerous physical systems where wave propagation or oscillations depend heavily on their historical states. As these applications scale in physical complexity, efficient parallel computing techniques become essential; however, developing highly scalable parallel solvers for delayed PDEs remains a significant computational challenge. To address this gap, this study extends the Dirichlet-Neumann Waveform Relaxation (DNWR) method to a multi-subdomain framework explicitly designed for solving time-delayed hyperbolic PDEs in parallel. We advance the underlying mathematical framework and provide a rigorous convergence analysis for both one-dimensional and two-dimensional spatial configurations. A central theoretical contribution of this work is the establishment of finite-step convergence for the multi-domain DNWR algorithm. These theoretical guarantees are firmly corroborated through comprehensive numerical experiments. Furthermore, a detailed comparative analysis against alternative domain decomposition techniques namely NNWR, OSWR, and Classical SWR, highlight the advantages of the proposed method. Finally, this approach provides a robust, highly parallelizable solver capable of efficiently handling complex hyperbolic PDEs with time delay.
ER -