Numerical investigations of the compressible Navier-Stokes system

Bilal Al-Taki (1) , Kevin Atsou (2) , Jean-Jéróme Casanova (3) , Thierry Goudon (2) , Pauline Lafitte (4) , Frédéric Lagoutière (5, 6) , Sebastian Minjeaud (2)

1 Peking University [Beijing]
2 COFFEE - COmplex Flows For Energy and Environment
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
5 ICJ - Institut Camille Jordan
6 MMCS - Modélisation mathématique, calcul scientifique

In this paper we write, analyze and experimentally compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: • a staggered scheme based on the Rusanov one for the inviscid (Euler) system, • a staggered pseudo-Lagrangian scheme in which the mesh "follows" the fluid, • the Eulerian projection (on a fixed mesh) of the preceding scheme. All these schemes only involve the resolution of linear systems (all the nonlinear terms are solved in an explicit way). We propose numerical illustrations of their behaviors on particular solutions in which the density has discontinuities (hereafter called Hoff solutions). We show that the three schemes seem to converge to the same solutions, and we compare the evolution of the amplitude of the discontinuity of the numerical solution (with the pseudo-Lagrangian scheme) with the one predicted by Hoff and observe a good agreement.

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]; Systèmes dynamiques [math.DS]; Analyse numérique [math.NA]

Bilal Al-Taki, Kevin Atsou, Jean-Jéróme Casanova, Thierry Goudon, Pauline Lafitte, et al.. Numerical investigations of the compressible Navier-Stokes system. ESAIM: Proceedings and Surveys, 2021, CEMRACS 2019 - Geophysical Fluids, Gravity Flows, 70, pp.1 - 13. ⟨10.1051/proc/202107001⟩. ⟨hal-03314984⟩