The method of freezing as a new tool for nonlinear reduced basis approximation of parameterized evolution equations

TL;DR

This paper introduces a novel nonlinear approximation method for parameterized evolution equations using a freezing technique, which improves efficiency and handles hyperbolic regimes with moving discontinuities.

Contribution

It develops a new approach that reformulates the problem via group decomposition and algebraic constraints, enabling efficient reduced basis approximation for complex nonlinear problems.

Findings

Significantly improved performance over non-freezing schemes

Effective handling of hyperbolic regimes with moving discontinuities

Feasible online evaluation of the reduced scheme

Abstract

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.