Consistent Internal Energy Based Schemes for the Compressible Euler Equations

Thierry Gallouët (1) , Raphaele Herbin (1) , J.-C Latché (2) , N Therme (3)

1 I2M - Institut de Mathématiques de Marseille
2 IRSN/PSN-RES/SA2I - Service des Agressions Internes et des risques Industriels
3 LMJL - Laboratoire de Mathématiques Jean Leray

Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the in- ternal energy equation, with corrective terms to ensure the correct cap- ture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either struc- tured meshes or general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms; these may be either based on semi-implicit pressure correction techniques or seg- regated in such a way that only explicit steps are involved (referred to hereafter as ”explicit” variants). In order to ensure the positivity of the density, the internal energy and the pressure, the discrete convection op- erators for the mass and internal energy balance equations are carefully designed; they use an upwind technique with respect to the material velocity only. The construction of the fluxes thus does not need any Rie- mann or approximate Riemann solver, and yields easily implementable algorithms. The stability is obtained without restriction on the time step for the pressure correction scheme and under a CFL-like condition for explicit variants: preservation of the integral of the total energy over the computational domain, and positivity of the density and the internal energy. The semi-implicit first-order upwind scheme satisfies a local discrete entropy inequality. If a MUSCL-like scheme is used in order to limit the scheme diffusion, then a weaker property holds: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L∞ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the mo- mentum balance. Still for the explicit scheme, with the above-mentioned MUSCL-like scheme, the same result only holds if the ratio of the time to the space step tends to zero.

Mots clés: Compressible Euler equations; internal energy; finite volume; staggered mesh; consistency; entropy estimates; compressible flows; Euler equations; pressure correction; segregated algorithms; Compressible Euler equations

Domaines: Analyse numérique [math.NA]

Thierry Gallouët, Raphaele Herbin, J.-C Latché, N Therme. Consistent Internal Energy Based Schemes for the Compressible Euler Equations. David Greiner, Maria Isabel Asensio, Rafael Montenegro. Numerical Simulation in Physics and Engineering: Trends and Applications, 1st ed. 2021 Lecture Notes of the XVIII ‘Jacques-Louis Lions' Spanish-French School, 24, Springer, 2021, SEMA SIMAI Springer Series. ⟨hal-01553699v2⟩