Nonlinear projection methods for multi-entropies Navier-Stokes systems

Abstract

Abstract. This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent entropies. Various models for complex compressible materials typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the generic impossibility of recasting equivalently the present system in full conservation form. Classical finite volume methods are shown to grossly fail in the capture of viscous shock solutions that are of primary interest in the present work. To enforce for validity a set of generalized jump conditions that we introduce, we propose a systematic and effective correction procedure, the so-called nonlinear projection method, and prove that it preserves all the stability properties satisfied by suitable Godunov-type methods. Numerical experiments assess the relevance of the method when exhibiting approximate solutions in close agreement with exact solutions.

A ‘well‐balanced’ finite volume scheme for blood flow simulation

“…Mostly [1,20], equations are in fact written in a non-conservative form. With non-conservative schemes, we have consistency defaults and shocks are not uniquely defined (among others [54][55][56]). Furthermore, the discharge Q is not conserved.…”

Why many theories of shock waves are necessary: kinetic relations for non-conservative systems

“…The identity (5.12) shows in addition that the mapping a → Λ 0 (a) can be built as soon as the jump in the last specific entropy s R N − s N L is known. This evaluation can be performed numerically; see, for instance, [8].…”

Time-Implicit Approximation of the Multipressure Gas Dynamics Equations in Several Space Dimensions

“…This explains the unacceptable errors generally observed between the exact and the numerical solutions after several numerical time steps. We refer the reader to [4], [12], [13], [14] for several illustrations. Note that by (2.4) we have…”

“…Let us stress that even the original Godunov method fails in producing good numerical results. This failure has been analyzed and cured in several contributions by Berthon-Coquel [2], [4], Chalons-Coquel [12], [13], [14], [16].…”

“…We begin by showing that classical methods like Godunov's scheme fail in providing numerical solutions in good agreement with exact ones. We proceed by a brief review of our previous works [4], [12] and [13] in which several relevant explicit in time numerical strategies are proposed. At last, we mention the main difficulties related to the extension of these strategies to an implicit in time framework, and give the basis of the developments proposed in the next section.…”