Modeling Advection-Dominated Flows with Space-Local Reduced-Order Models

Abstract

Reduced-order models (ROMs) are often used to accelerate the simulation of large physical systems. However, traditional ROM techniques, such as those based on proper orthogonal decomposition (POD), often struggle with advection-dominated flows due to the slow singular value decay. This results in high computational costs and potential instabilities. This paper proposes a novel approach using space-local POD to address the challenges arising from the slow singular value decay. Instead of global basis functions, our method employs local basis functions that are applied across the domain, analogous to the finite element method, but with a data-driven basis. By dividing the domain into subdomains and applying the space-local POD, we achieve a representation that is sparse and that generalizes better outside the training regime. This allows the use of a larger number of basis functions compared to standard POD, without prohibitive computational costs. To ensure smoothness across subdomain boundaries, we introduce overlapping subdomains inspired by the partition of unity method. Our approach is validated through simulations of the 1D and 2D advection equation. We demonstrate that using our space-local approach we obtain a ROM that generalizes better to flow conditions which are not part of the training data. In addition, we show that the constructed ROM inherits the energy conservation and non-linear stability properties from the full-order model. Finally, we find that using a space-local ROM allows for larger time steps.