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

TL;DR

The paper tackles stable model reduction for one-dimensional nonlinear hyperbolic systems by extending entropy-stable ROMs to nonlinear manifolds and introducing tangent-space enrichment (TSE) and rational quadratic manifolds to better capture shocks.A nonlinear manifold Galerkin ROM is developed where the projection is performed on the tangent space of the manifold and evaluated at an entropy-projected state, ensuring semi-discrete entropy stability analogous to the full-order model.Tangent space enrichment lifts the manifold along entropy directions to keep the entropy-projection error small, while rational quadratic manifolds provide a more accurate, pole-free parameterization that handles sharp gradients without spurious oscillations.Numerical experiments on the inviscid Burgers, shallow water, and compressible Euler equations demonstrate that the proposed ES-ROMs preserve entropy correctly and achieve higher reconstruction accuracy with smaller reduced dimensions compared to linear and quadratic manifold approaches.

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.

Figures (26)

• Figure 1: Flowchart of possible entropy stable ROM approaches, our method is indicated in red, $u$ are the conserved variables and $\eta$ are the alternate variables used in kalashnikovastable. FOM: full order model, ROM: reduced order model.
• Figure 2: A visualization of the ROM construction. The entropy variables $\bm{\eta}_r$ are projected on the tangent space $T_{u_r}\mathcal{M}$ to obtain $\tilde{\bm{\eta}}_r$. A new state $\bm{u}_h$ (not necessarily on the reduced manifold) is found with entropy variables $\bm{\eta}_h$ such that $\tilde{\bm{\eta}}_r = \bm{\eta}_h$. We set $\tilde{\bm{u}}_r = \bm{u}_h$ and project $-\Delta_v \bm{f}_h(\tilde{\bm{u}}_r)$ orthogonally on the tangent space to complete the ROM.
• Figure 3: A visualization of tangent space enrichment. The dark purple curve is the original manifold $\mathcal{M}$, the light purple region is a section of the enriched manifold $\mathcal{M}^*$. $\mathcal{M}^*$ is constructed by lifting points from $\varphi(\bm{a})$ along lines in the direction of $\bm{\eta}(\varphi(\bm{a}))$.
• Figure 4: Normalized singular values of the inviscid Burgers equation data.
• Figure 5: Original data and reconstructions for $r=15$ in space-time plots.

Theorems & Definitions (1)

• Definition 1: Entropy-conservative numerical flux