Dirichlet-Neumann波形松弛方法用于多子域中带时滞的双曲型偏微分方程

中文摘要 AI 辅助

带时滞的双曲型偏微分方程（PDEs）是模拟众多物理系统的重要数学工具，这些系统中波的传播或振荡强烈依赖于其历史状态。随着这些应用在物理复杂性上的扩展，高效的并行计算技术变得至关重要；然而，为带时滞的偏微分方程开发高度可扩展的并行求解器仍然是一个重大的计算挑战。为解决这一空白，本研究将Dirichlet-Neumann波形松弛（DNWR）方法扩展到一个多子域框架，专门用于并行求解带时滞的双曲型偏微分方程。我们推进了基础数学框架，并为一维和二维空间配置提供了严格的收敛性分析。本工作的一个核心理论贡献是建立了多域DNWR算法的有限步收敛性。这些理论保证通过全面的数值实验得到了有力证实。此外，与替代域分解技术（即NNWR、OSWR和经典SWR）的详细比较分析突出了所提出方法的优势。最后，该方法提供了一个稳健、高度可并行化的求解器，能够高效处理带时滞的复杂双曲型偏微分方程。

英文摘要

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.

发表机构

• School of Basic Sciences, IIT Bhubaneswar（IIT布巴内斯瓦尔基础科学学院）

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