用域分解方法高效耦合降阶模型，提升非线性问题求解速度与精度。
The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models
- 采用施瓦茨交替法，通过边界条件迭代传递信息实现子域降阶模型耦合。
- 在非重叠分解下使用狄利克雷-狄利克雷边界条件，实现稳定高精度求解。
- 可实现比全阶模型快100倍、比整体模型快中等程度的加速效果。
本文提出并评估了一种基于施瓦茨交替法的框架，用于耦合子域局部的投影型降阶模型（PROMs），该方法对空间域进行域分解。通过迭代求解一系列子域局部问题，信息通过传输边界条件在子域间传播，最终获得全域解。针对三个二维非线性双曲问题——浅水方程、伯格斯方程和可压缩欧拉方程，研究了施瓦茨交替法的新方向，以提升方法效率与灵活性。对于单元中心有限体积离散及非重叠域分解，结果表明：采用狄利克雷-狄利克雷（而非罗宾-罗宾或交替狄利克雷-诺伊曼）传输边界条件，仍可获得稳定且准确的耦合模型。此外，还探索了在耦合超降阶PROM时边界采样策略的影响。数值结果表明，该方法通过域分解实现了模型的空间局域化，能将降阶模型精度提升，并相较等效耦合全阶模型实现高达两个数量级的加速，同时较类似整体解法获得中等程度加速。
原文摘要 · Abstract (English)
This paper presents and evaluates a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method following a domain decomposition (DD) of the spatial domain on which a given problem of interest is posed. In this approach, the solution on the full domain is obtained via an iterative process in which a sequence of subdomain-local problems are solved, with information propagating between subdomains through transmission boundary conditions (BCs). We explore several new directions involving the Schwarz alternating method aimed at maximizing the method's efficiency and flexibility, and demonstrate it on three challenging two-dimensional nonlinear hyperbolic problems: the shallow water equations, Burgers' equation, and the compressible Euler equations. We demonstrate that, for a cell-centered finite volume discretization and a non-overlapping DD, it is possible to obtain a stable and accurate coupled model utilizing Dirichlet-Dirichlet (rather than Robin-Robin or alternating Dirichlet-Neumann) transmission BCs on the subdomain boundaries. We additionally explore the impact of boundary sampling when utilizing the Schwarz alternating method to couple subdomain-local hyper-reduced PROMs. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy by enabling the spatial localization of these models via domain decomposition, and achieve up to two orders of magnitude speedup over equivalent coupled full order model solutions and moderate speedups over analogous monolithic solutions.
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