Robust Model Reduction Of Hyperbolic Problems by $L^1$-norm Minimization and Dictionary Approximation

TL;DR

This paper introduces a new model reduction technique for hyperbolic equations that uses a dictionary of solutions and $L^1$-norm minimization to accurately approximate solutions with shocks and discontinuities.

Contribution

It presents a novel framework combining dictionary-based approximation with $L^1$-norm minimization tailored for hyperbolic problems, including efficient algorithms for residual minimization.

Findings

Accurately approximates solutions with shocks and discontinuities.

Requires only a few modes for accurate results.

Produces physically acceptable, oscillation-free solutions.

Abstract

We propose a novel model reduction approach for the approximation of non linear hyperbolic equations in the scalar and the system cases. The approach relies on an offline computation of a dictionary of solutions together with an online $L^1$-norm minimization of the residual. It is shown why this is a natural framework for hyperbolic problems and tested on nonlinear problems such as Burgers' equation and the one-dimensional Euler equations involving shocks and discontinuities. Efficient algorithms are presented for the computation of the $L^1$-norm minimizer, both in the cases of linear and nonlinear residuals. Results indicate that the method has the potential of being accurate when involving only very few modes, generating physically acceptable, oscillation-free, solutions.