On snapshot-based model reduction under compatibility conditions for a nonlinear flow problem on networks

TL;DR

This paper develops a structure-preserving, snapshot-based model reduction method for nonlinear flow problems on networks, ensuring compatibility conditions that maintain physical properties and improve efficiency.

Contribution

It introduces a novel reduction approach combining variational Galerkin approximation with quadrature-type complexity reduction under compatibility constraints, ensuring structure preservation.

Findings

Reduced models are locally mass conservative.

Models inherit energy bounds and port-Hamiltonian structure.

The method achieves optimal snapshot data approximation.

Abstract

This paper is on the construction of structure-preserving, online-efficient reduced models for the barotropic Euler equations with a friction term on networks. The nonlinear flow problem finds broad application in the context of gas distribution networks. We propose a snapshot-based reduction approach that consists of a mixed variational Galerkin approximation combined with quadrature-type complexity reduction. Its main feature is that certain compatibility conditions are assured during the training phase, which make our approach structure-preserving. The resulting reduced models are locally mass conservative and inherit an energy-bound and port-Hamiltonian structure. We also derive a well-posedness result for them. In the training phase, the compatibility conditions pose challenges, we face constrained data approximation problems as opposed to the unconstrained training problems in the conventional reduction methods. The training of our model order reduction consists of a principal component analysis under a compatibility constraint and, notably, yields reduced models that fulfill an optimality condition for the snapshot data. The training of our quadrature-type complexity reduction involves a semi-definite program with combinatorial aspects, which we approach by a greedy procedure.