Registration-based model reduction of parameterized two-dimensional conservation laws

TL;DR

This paper introduces a nonlinear registration-based model reduction method for efficiently solving parameterized 2D conservation laws with discontinuities, combining feature tracking and hyper-reduction techniques.

Contribution

It develops a general registration procedure for 2D hyperbolic systems and proposes a multi-fidelity approach to reduce offline computational costs.

Findings

Effective in handling parameter-dependent discontinuities.

Successfully applied to supersonic flow past a bump.

Reduces offline costs with multi-fidelity strategy.

Abstract

We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping $\Phi$ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.