Invariant domains and first-order continuous finite element approximation for hyperbolic systems

TL;DR

This paper introduces a finite element numerical method for hyperbolic systems that ensures solutions stay within invariant sets and satisfy entropy conditions, using artificial dissipation and CFL constraints.

Contribution

It extends invariant domain preserving techniques to continuous finite elements for hyperbolic systems in multiple dimensions, ensuring stability and entropy compliance.

Findings

Invariant domain property holds under CFL condition.

Method satisfies discrete entropy inequalities.

Can be extended to high-order in time with SSP algorithms.

Abstract

We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant sets containing the initial data is an invariant domain for the method. The invariant domain property is proved for any hyperbolic system provided a CFL condition holds. The solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system. The method is formally first-order accurate in space and can be made high-order in time by using Strong Stability Preserving algorithms. This technique extends to continuous finite elements the work of \cite{Hoff_1979,Hoff_1985}, and \cite{Frid_2001}.