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

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.

On the Stability of Robust Dynamical Low-Rank Approximations for Hyperbolic Problems

“…However, the augmented and midpoint BUG integrators enable preservation independent of the time step size. Note, however, that for most kinetic problems, structure preservation can be enforced by other means via further basis augmentations [16,22], or micro-macro decompositions [34].…”

A rank-adaptive robust integrator for dynamical low-rank approximation

“…Dynamical low-rank approximation of time-dependent matrices [37] has proven to be an efficient model order reduction technique for applications from widely varying fields including plasma physics [24,26,20,8,22,14,27,15,23,56], radiation transport [49,16,47,41,48,57,21,4], radiation therapy [40], chemical kinetics [32,50,25], wave propagation [31,58], kinetic shallow water models [38], uncertainty quantification [51,3,28,43,44,46,39,33,17,2], and machine learning [54,59,52,53]. These problems can be written as a prohibitively large matrix differential equation for A(t) ∈ R m×n , .…”