Dissipative issue of high-order shock capturing schemes with non-convex equations of state

• Published in: Journal of computational physics 2009-02, Vol.228 (3), p.833-860
• Main Authors: Heuzé, Olivier, Jaouen, Stéphane, Jourdren, Hervé
• Format: Article
• Language: English
• Subjects: Artificial viscosity; Benchmarks; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Composite waves; COMPRESSIBLE FLOW; Compressible fluid dynamics; Computational techniques; Dissipation; EQUATIONS OF STATE; Exact sciences and technology; FLUID MECHANICS; Flux limiters; Fundamental derivative; Godunov-type methods; High-order schemes; Initial value problems; ISENTROPIC PROCESSES; LAGRANGIAN FUNCTION; Mathematical methods in physics; Mathematical models; MATHEMATICAL SOLUTIONS; Non-convex equation of state; Physics; Rarefaction; Riemann problem; Shock capturing; SHOCK WAVES; Sonics; vNR-type methods
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2008.10.005

It is well known that, closed with a non-convex equation of state (EOS), the Riemann problem for the Euler equations allows non-standard waves, such as split shocks, sonic isentropic compressions or rarefaction shocks, to occur. Loss of convexity then leads to non-uniqueness of entropic or Lax solutions, which can only be resolved via the Liu-Oleinik criterion (equivalent to the existence of viscous profiles for all admissible shock waves). This suggests that in order to capture the physical solution, a numerical scheme must provide an appropriate level of dissipation. A legitimate question then concerns the ability of high-order shock capturing schemes to naturally select such a solution. To investigate this question and evaluate modern as well as future high-order numerical schemes, there is therefore a crucial need for well-documented benchmarks. A thermodynamically consistent C ∞ non-convex EOS that can be easily introduced in Eulerian as well as Lagrangian hydrocodes for test purposes is here proposed, along with a reference solution for an initial value problem exhibiting a complex composite wave pattern (the Bizarrium test problem). Two standard Lagrangian numerical approaches, both based on a finite volume method, are then reviewed (vNR and Godunov-type schemes) and evaluated on this Riemann problem. In particular, a complete description of several state-of-the-art high-order Godunov-type schemes applicable to general EOSs is provided. We show that this particular test problem reveals quite severe when working on high-order schemes, and recommend it as a benchmark for devising new limiters and/or next-generation highly accurate schemes.