Consistent Internal Energy Based Schemes for the Compressible Euler Equations

Autor: Nicolas Therme, Raphaèle Herbin, Thierry Gallouët, Jean-Claude Latché
Přispěvatelé: Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Radioprotection et de Sûreté Nucléaire (IRSN), Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - Faculté des Sciences et des Techniques, Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), David Greiner, Maria Isabel Asensio, Rafael Montenegro, PSN-RES/SA2I, Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Service des Agressions Internes et des risques Industriels (IRSN/PSN-RES/SA2I), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST)
Rok vydání: 2021
Předmět:
Zdroj: Numerical Simulation in Physics and Engineering: Trends and Applications ISBN: 9783030625429
Numerical Simulation in Physics and Engineering: Trends and Applications, 1st ed. 2021 Lecture Notes of the XVIII ‘Jacques-Louis Lions' Spanish-French School
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
DOI: 10.1007/978-3-030-62543-6_3
Popis: International audience; 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.
Databáze: OpenAIRE
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