A class of asymptotic preserving numerical schemes for low Mach number flows
Speaker: Khaled Saleh

Time: 9h30, Tuesday, January 23, 2018
Room 4, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi

Abstract: Since seminal papers published in the middle of the sixties, low-order staggered schemes for incompressible flow computations have received a considerable attention. The staggered discretization is a space structured or unstructured discretization where the scalar unknowns (the pressure) are located at the cell centers while the vector unknowns (the velocity) are located at the cells faces. This interest is essentially motivated by the fact that they combine a low computational cost with the so-called inf-sup or LB stability condition, which prevents from the odd-even decoupling of the pressure.

For several years, an important effort has been dedicated to the extension of staggered numerical schemes for the approximation of compressible flows. Usually expected properties have been proven for these new schemes: existence of the numerical solution, preservation of the admissible phase space, stability, entropy inequalities, consistency in the sens of Lax and Wendroff. Besides their efficiency and simplicity (the numerical fluxes are easily computed), these schemes have the following remarkable property: if the numerical density is constant, they boil down to the standard staggered algorithm for incompressible flows, which is known to be stable and efficient. In particular, as for the incompressible case, the staggered discretization ensures the control of the L^2 norm of the pressure through of the control of the H^{-1} norm of its gradient (thanks to the inf-sup stability condition).

In this talk, I will focus on the barotropic compressible Navier-Stokes equations. I will prove that, as the Mach number tends to zero, the solution of the implicit staggered scheme for these equations converges towards the solution of the standard staggered scheme for the incompressible Navier-Stokes equations. In particular, the numerical density tends towards a constant as the Mach number tends to zero. Such a result follows from a similar analysis to that of  Lions and Masmoudi (1998) at the continuous level for weak solutions of thebarotropic compressible Navier-Stokes equations. It extends to other time discretizations such as the so called pressure-correction scheme (adapted to compressible models). It is these schemes that are used in practice in the industrial codes such as P2REMICS. A code developed by the IRSN (french Institut for Radioprotection and Nuclear Safety) for the simulation of deflagrations.


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