Hybrid Surrogate Models: Circumventing Gibbs Phenomenon for Partial Differential Equations with Finite Shock-Type Discontinuities

TL;DR

This paper presents Hybrid Surrogate Models that combine polynomials and Heavyside functions to effectively approximate PDE solutions with shocks, overcoming Gibbs phenomenon limitations.

Contribution

The introduction of Hybrid Surrogate Models that integrate polynomial and Heavyside functions for improved PDE approximation with discontinuities.

Findings

HSMs accurately approximate shock solutions in PDEs.

HSMs effectively identify shock positions and magnitudes.

Numerical experiments confirm superior approximation over traditional methods.

Abstract

We introduce the concept of Hybrid Surrogate Models (HSMs) -- combining multivariate polynomials with Heavyside functions -- as approximates of functions with finitely many jump discontinuities. We exploit the HSMs for formulating a variational optimization approach, solving non-regular partial differential equations (PDEs) with non-continuous shock-type solutions. The HSM technique simultaneously obtains a parametrization of the position and the height of the shocks as well as the solution of the PDE. We show that the HSM technique circumvents the notorious Gibbs phenomenon, which limits the accuracy that classic numerical methods reach. Numerical experiments, addressing linear and non-linearly propagating shocks, demonstrate the strong approximation power of the HSM technique.