The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models

TL;DR

This paper develops a Schwarz-based framework for coupling projection-based reduced order models using domain decomposition, demonstrating improved accuracy and significant computational speedups on complex nonlinear hyperbolic problems.

Contribution

It introduces a novel coupling approach with boundary condition strategies and sampling techniques that enhance PROM efficiency and flexibility in nonlinear hyperbolic PDEs.

Findings

Achieves stable, accurate coupling with Dirichlet-Dirichlet boundary conditions.

Provides up to two orders of magnitude speedup over full order models.

Shows potential for improved PROM accuracy via domain decomposition.

Abstract

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.