Entropy-Stable Model Reduction of One-Dimensional Hyperbolic Systems using Rational Quadratic Manifolds

TL;DR

This paper introduces a new entropy-stable reduced order modeling approach for one-dimensional hyperbolic systems that uses rational quadratic manifolds to improve accuracy while preserving physical entropy conditions.

Contribution

It generalizes entropy-stable ROMs to nonlinear manifolds, especially rational quadratic ones, ensuring entropy inequalities are satisfied and enhancing model accuracy.

Findings

ROMs preserve entropy stability on nonlinear manifolds.

Rational quadratic manifolds improve approximation accuracy.

Numerical experiments confirm enhanced structure-preserving properties.

Abstract

In this work we propose a novel method to ensure important entropy inequalities are satisfied semi-discretely when constructing reduced order models (ROMs) on nonlinear reduced manifolds. We are in particular interested in ROMs of systems of nonlinear hyperbolic conservation laws. The so-called entropy stability property endows the semi-discrete ROMs with physically admissible behaviour. The method generalizes earlier results on entropy-stable ROMs constructed on linear spaces. The ROM works by evaluating the projected system on a well-chosen approximation of the state that ensures entropy stability. To ensure accuracy of the ROM after this approximation we locally enrich the tangent space of the reduced manifold with important quantities. Using numerical experiments on some well-known equations (the inviscid Burgers equation, shallow water equations and compressible Euler equations) we show the improved structure-preserving properties of our ROM compared to standard approaches and that our approximations have minimal impact on the accuracy of the ROM. We additionally generalize the recently proposed polynomial reduced manifolds to rational polynomial manifolds and show that this leads to an increase in accuracy for our experiments.