A second order anti-diffusive Lagrange-remap scheme for two-component flows

Abstract

Abstract. We build a non-dissipative second order algorithm for the approximate resolution of the one-dimensional Euler system of compressible gas dynamics with two components. The considered model was proposed in [1]. The algorithm is based on [8] which deals with a non-dissipative first order resolution in Lagrange-remap formalism. In the present paper we describe, in the same framework, an algorithm that is second order accurate in time and space, and that preserves sharp interfaces. Numerical results reported at the end of the paper are very encouraging, showing the interest of the second order accuracy for genuinely non-linear waves.Résumé. Nous construisons un algorithme d'ordre deux et non dissipatif pour la résolution approchée deséquations d'Euler de la dynamique des gaz compressiblesà deux constituants en dimension un. Le modèle que nous considérons est celuià cinqéquations proposé et analysé dans [1]. L'algorithme est basé sur [8] qui proposait une résolution approchéeà l'ordre un et non dissipative au moyen d'un splitting de type Lagrange-projection. Dans le présent article, nous décrivons, dans le même formalisme, un algorithme d'ordre deux en temps et en espace, qui préserve des interfaces "parfaites" entre les constituants. Les résultats numériques rapportésà la fin de l'article sont très encourageants ; ils montrent clairement les avantages d'un schéma d'ordre deux pour les ondes vraiment non linéaires.

An anti-diffusive Lagrange-Remap scheme for multi-material compressible flows with an arbitrary number of components

“…In particular, they demonstrate the great advantage of the andi-diffusive scheme which is simple to implement and very efficient especially to accurately treat the interface. Extension to the second order in space using MUSCL reconstruction is in progress adapting [4], especially to improve the nonlinear waves resolution. …”

“…In this section, we present an anti-diffusive Lagrange-Remap method [8,10] for our m-component system (2) adapted from the m = 2 case [4,12]. This discretization relies on a two-step splitting that decouples the acoustic effects taken into account by Lagrange step from the transport that is approximated within the Remap step.…”

“…In order to limit the numerical diffusion of the color functions, we adopt the strategy proposed by [5,7] and applied in [4,11,12]: for k = 1, . .…”

Simulation of sharp interface multi-material flows involving an arbitrary number of components through an extended five-equation model

“…The system of equations (8) will be referred to as the m-component model. This system is quasi-conservative since the equation (8d) is not conservative.…”

“…This system is quasi-conservative since the equation (8d) is not conservative. We shall see in Appendix A that this is not an issue: the non-conservative product u · ∇Z k is well-defined and one can recast system (8) into an equivalent fully-conservative formulation.…”