First-order greedy invariant-domain preserving approximation for hyperbolic problems: scalar conservation laws, and p-system

TL;DR

This paper introduces a novel method for estimating artificial viscosity in hyperbolic system approximations, ensuring invariant-domain preservation and entropy compliance by focusing on minimum wave speeds rather than maximum wave speeds.

Contribution

It proposes a new approach to estimate artificial viscosity using minimum wave speeds, improving invariant-domain and entropy inequality preservation in numerical methods for hyperbolic problems.

Findings

Eliminates non-essential fast waves from artificial viscosity estimation

Ensures invariant-domain properties and entropy inequalities are satisfied

Provides a more accurate and stable numerical approximation method

Abstract

The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities.