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

TL;DR

The paper addresses the challenge of efficiently simulating advection-dominated flows with reduced-order models by introducing space-local POD, which uses a common local basis across subdomains and overlapping regions to achieve sparsity and smoother transitions. The authors derive a Galerkin ROM that preserves the energy conservation of the full-order model and demonstrate improved generalization to unseen flow regimes, as well as the potential for larger time steps. Through 1D and 2D advection (and preliminary Navier–Stokes tests), they show that space-local ROMs markedly outperform global POD in extrapolation, while LO-POD provides smoother solutions at a modest computational cost. The approach offers a practical route to stable, efficient ROMs for advection-dominated systems and sets the stage for extensions to more complex turbulence problems.

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.

Figures (19)

• Figure 1: A Gaussian wave discretized by a finite difference scheme represented by a local basis of box functions for $I=3$ subdomains and $J = 8$ points per subdomain. The edges of the subdomains are indicated by the vertical grey lines.
• Figure 2: Schematic representation of the snapshot matrix $\mathbf{X}$ being reshaped into the snapshot matrix $\mathbf{X}_\ell$ for $I = 3$.
• Figure 3: and LO-POD representation of a Gaussian wave. The data used to obtain the local basis is explained in Section \ref{['sec:results']}. The depicted snapshot is part of the snapshot matrix used to obtain the basis. The edges of the subdomains are indicated by the vertical grey lines. Only part of the domain $\Omega = [0,2\pi)$ is shown.
• Figure 4: Kernels $k_i$ for a subdivision of the periodic domain into five overlapping subdomains of equal size.
• Figure 5: Reference simulation of a Gaussian wave being advected throughout the domain. White bars separate the training data from the validation data and the validation data from the extrapolation data, respectively.