Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

TL;DR

This paper introduces a novel model reduction method for parameterized hyperbolic PDEs that effectively handles shocks and discontinuities through space-time registration, adaptive data compression, and hyper-reduction techniques.

Contribution

It develops an innovative space-time registration-based model reduction framework specifically designed for hyperbolic PDEs with discontinuities, improving computational efficiency and reliability.

Findings

Successfully applied to Burgers and shallow water models.

Demonstrates significant speed-up in online computations.

Effectively captures shock and contact discontinuities.

Abstract

We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov-Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.