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Abstract
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.
Highlights
We are interested in the simulation of the following Compressible Navier-Stokes system∂tρ + ∂x(ρu) = 0, (1)∂t(ρu) + ∂x(ρu2) + ∂xp − ∂x(μ∂xu) = 0.In (1), the unknowns (t, x) → ρ(t, x) and (t, x) → u(t, x) stand respectively for the density and the velocity of a fluid
The Eulerian projection of the preceding scheme. All these schemes only involve the resolution of linear systems
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 one predicted by Hoff and observe a good agreement
Summary
We are interested in the simulation of the following Compressible Navier-Stokes system. For further results and references about the existence theory for the compressible Navier-Stokes system, we refer the reader to [6, 19] and the reference books [7, 22] It is worth mentioning the recent breakthrough [4] which deals with more intricate pressure laws and introduce new compactness arguments, and [5, 25, 27] for results on the case of density-dependent viscosities.
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