Invariant domains preserving ALE approximation of hyperbolic systems with continuous finite elements

TL;DR

This paper introduces a conservative ALE method for hyperbolic systems that preserves invariant domains, is explicit, continuous finite element-based, and uniquely defines artificial viscosity independently of mesh geometry.

Contribution

It presents a novel invariant domain preserving ALE scheme with unambiguous artificial viscosity, serving as a foundation for higher-order spatial methods.

Findings

Method is conservative and invariant domain preserving.

Artificial viscosity is mesh-independent and parameter-free.

Suitable for extension to higher-order methods.

Abstract

A conservative invariant domain preserving Arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems is introduced. The method is explicit in time, works with continuous finite elements and is first-order accurate in space. One originality of the present work is that the artificial viscosity is unambiguously defined irrespective of the mesh geometry/anisotropy and does not depend on any ad hoc parameter. The proposed method is meant to be a stepping stone for the construction of higher-order methods in space by using appropriate limitation techniques.