An Arbitrary Lagrangian-Eulerian strategy to solve compressible fluid flows

TL;DR: This paper proposes some schemes for each step of the computation of bi-dimensional compressible fluid dynamics written in Lagrangian frame and proposes also a local way to impose a discrete maximum principle for arbitrary second order limiter (or high order method) without any repair process.

Abstract: In this paper, we consider an Arbitrary Lagrangian Eulerian (ALE) alternative to the computation of bi-dimensional compressible fluid dynamics written in Lagrangian frame. In this formulation, the conservation laws are solved on a mesh moving at the speed of the flow. In the context of multi-material flows, this Lagrangian description allows the exact preservation of the interface between different species. Originally, Lagrangian schemes considered a staggered discretization of unknowns (density, velocity, energy), the velocity is located at nodes and density, pressure are cell centered. Recently, new schemes have been designed as a full centered version . These two schemes can be seen as an extension of Godunov scheme to fully multi-dimensional flux (at nodes). Each one verifies a local discrete vertex centered entropy inequality. Despite all efforts to obtain robust simulations for Inertial Confinement Fusion (ICF) computations, nowadays, the pure Lagrangian frame mode does not ensure a good cell quality during the calculation: for example non convex cell and/or tangled cell may appear. A commonly used alternative is to consider an ALE formulation. A pure Lagrangian phase is followed by a two step process: the rezoning of the Lagrangian mesh followed by a remapping step. The aim of this paper is to propose some schemes for each step. In the first one, we will introduce a strategy that controls the quality of the rezoned mesh for arbitrary polygonal conformal mesh under the constraint of being close to the Lagrangian grid. For the second one, we propose an extension of swept volume based remapping method. We propose also a local way to impose a discrete maximum principle (local bound preservation) for arbitrary second order limiter (or high order method) without any repair process.