Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation

TL;DR

This work tackles efficient, mass-conserving MOR for hyperbolic shallow water moment equations by separating conservative macro dynamics from non-conservative microscopic corrections. It develops two reduction strategies, POD-Galerkin with an offline-online workflow and dynamical low-rank approximation with online basis updates, both preserving mass and consistency with the SWE limit as microscopic structure vanishes. The macro-micro decomposition enables accurate reduced models for the micro moments while keeping the macroscopic water height and momentum intact, and numerical experiments (dam-break, smooth wave, square-root profile) demonstrate high accuracy with small ranks and substantial speedups. The study reveals tradeoffs between offline costs and online adaptivity, with rank-adaptive DLRA offering robust performance for transport-dominated shallow water flows and suggesting avenues for integrating these approaches in higher-dimensional, parameter-rich settings.

Abstract

Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.

Figures (15)

• Figure 1: Concept of the macro-micro decomposition MOR approach for HSWME. The HSWME is decomposed in macroscopic variables $\textrm{\boldmath${u}$}$ and microscopic variables $\textrm{\boldmath${v}$}$. While the macroscopic variables are computed with a standard conservative scheme, the microscopic variables are treated by MOR techniques DLRA or POD-Galerkin.
• Figure 2: Sampled snapshot data of the POD, shown for the water height component.
• Figure 3: Relative approximation error $\sum_i\Vert{\textrm{\boldmath${v}$}(x,t_i)-\tilde{\textrm{\boldmath${v}$}}(x,t_i)}\Vert_{L_2}$ of the snapshot data for space-moment $\tilde{\textrm{\boldmath${v}$}}(x,t)=\sum_k \alpha_k(t) \textrm{\boldmath${w}$}_k(x)$ and moment only basis $\tilde{\textrm{\boldmath${v}$}}(x,t) =\sum_{k=1}^r \hat{\alpha}_k(x,t) \textrm{\boldmath${w}$}_k$. The online error is given as $\Vert{\textrm{\boldmath${u}$}(x,t_\text{end})-\tilde{\textrm{\boldmath${u}$}}(x,t_\text{end})}\Vert_{L_2}$, where $\textrm{\boldmath${u}$}=(h, h u_m)$
• Figure 4: Water column test case comparison of macroscopic quantities water height $h$ (a) and momentum $h u_m$ (b) for full-order HSWME, SWE, POD-Galerkin, and DLRA using rank $r=5$. Both reduced models DLRA and POD-Galerkin yield indistinguishable results from the full-order model while the simple SWE model shows insufficient accuracy.
• Figure 5: Water column test case comparison of velocity profiles close to the domain center (a) and close to the shock wave (b) for full-order HSWME, SWE, POD-Galerkin, and DLRA using rank $r=5$. Both reduced models DLRA and POD-Galerkin also yield indistinguishable results from the full-order model while the simple SWE model shows insufficient accuracy and wrong propagation speeds.

Theorems & Definitions (4)

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