An Asymptotic-Preserving all-speed scheme for the Euler and Navier–Stokes equations

Abstract

We present an Asymptotic-Preserving 'all-speed' scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and Two-dimensional numerical results provide a validation of the Asymptotic-Preserving 'all-speed' properties.

Modelling of phase transitions in granular flows

“…Several ideas have been proposed to tackle this issue among which preconditioning methods (see [28]) which consist in multiplying the time derivatives by an appropriate matrix in order to alter the stiff eigenvalues of the system, and pressure correction methods, which extend to the compressible setting the projection techniques introduced by Chorin [8], Temam [27] in the incompressible framework (see for instance the works of Harlow, Amsden [14] or more recently the works of Herbin, Kheriji, Latché [16]). We are interested in this last category and more precisely in the recent techniques of implicit/explicit (IMEX) discretizations proposed for instance by Klein [18], Degond et al [10], Degond and Tang [13], Cordier, Degond, Kumbaro [9], Noelle et al [23]. These methods consist in splitting the pressure into a stiff and a non-stiff part, the first one being treated implicitly in time whereas the non-stiff part is treated explicitly.…”

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

“…The method derived and studied in this work belongs to the class of schemes called asymptotic preserving (AP). Such type of schemes have been developed in [4] and in [6,8] for the specific problem related to the compressible-incompressible passage. However, in [4] the method is based on a Lagrange projection method and on a splitting procedure that allows to decouple the acoustic and the transport phenomenons.…”

“…However, in [4] the method is based on a Lagrange projection method and on a splitting procedure that allows to decouple the acoustic and the transport phenomenons. While in [8,6], the authors split the pressure through the introduction of a numerical parameter which must be tuned depending on the problem in order for the scheme to work. Here, we use an alternative approach.…”

A Relaxation Scheme for the Simulation of Low Mach Number Flows

“…This way the correct numerical viscosity for each regime is recovered. The upwind scheme is derived as in [5], computing the interface values by using the characteristic variables v ± A 1/2 ψ of system (2). At first order we have:…”

“…Nevertheless, the upwind discretization is needed in presence of Mach numbers of order one, in order to introduce enough numerical viscosity. Since the adopted spatial discretization is a convex combination of these two, it is able to approximate flows in different regimes and thus the scheme can be seen as an "all-speed" scheme, as the ones proposed in [2,3]. Moreover, thanks to the absolute stability of implicit schemes, the CFL is not limited by the acoustic constraint, which becomes extremely demanding in low Mach regimes.…”