Invariant Domains Preserving Arbitrary Lagrangian Eulerian Approximation of Hyperbolic Systems with Continuous Finite Elements

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 original element 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.