Conservative model reduction for finite-volume models

TL;DR

This paper introduces a conservative model reduction technique for finite-volume models that guarantees conservation properties and improves accuracy over traditional nonconservative methods, using optimization-based approaches with hyper-reduction.

Contribution

It develops a novel optimization-based framework for conservative model reduction of finite-volume models, ensuring conservation and reducing errors compared to existing methods.

Findings

Ensures global conservation in reduced-order models.

Achieves lower state-space errors than standard Galerkin and LSPG methods.

Provides conditions for equivalence and error bounds for the proposed approaches.

Abstract

This work proposes a method for model reduction of finite-volume models that guarantees the resulting reduced-order model is conservative, thereby preserving the structure intrinsic to finite-volume discretizations. The proposed reduced-order models associate with optimization problems characterized by a minimum-residual objective function and nonlinear equality constraints that explicitly enforce conservation over subdomains. Conservative Galerkin projection arises from formulating this optimization problem at the time-continuous level, while conservative least-squares Petrov--Galerkin (LSPG) projection associates with a time-discrete formulation. We equip these approaches with hyper-reduction techniques in the case of nonlinear flux and source terms, and also provide approaches for handling infeasibility. In addition, we perform analyses that include deriving conditions under which conservative Galerkin and conservative LSPG are equivalent, as well as deriving a posteriori error bounds. Numerical experiments performed on a parameterized quasi-1D Euler equation demonstrate the ability of the proposed method to ensure not only global conservation, but also significantly lower state-space errors than nonconservative reduced-order models such as standard Galerkin and LSPG projection.