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

Dimarco, Giacomo; Loubère, Raphaël; Vignal, Marie-Hélène

doi:10.1137/16m1069274

Cited by 55 publications

(64 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Several AP-schemes were designed in the last years for various types of problems, including anisotropic elliptic [13,14] or parabolic [28] equations, Vlasov equation in the hydrodynamic regime [19] or drift-diffusion regime [10,24], Vlasov equation in the high-field limit [11,23], Euler equation in the low-Mach regime [15,16]. Briefly, an APscheme is a numerical scheme specially designed for singularly-perturbed problems P , containing some small parameter 1, and which enjoy the following properties (see commutative diagram 1):…”

Section: Introductionmentioning

confidence: 99%

Asymptotic-Preserving Scheme for the Resolution of Evolution Equations with Stiff Transport Terms

Fedele¹,

Negulescu²,

Possanner³

2019

Multiscale Model. Simul.

View full text Add to dashboard Cite

We develop an asymptotic-preserving scheme to solve evolution problems containing stiff transport terms. This scheme is based to a micro-macro decomposition of the unknown, coupled with a stabilization procedure. The numerical method is applied to the Vlasov equation in the gyrokinetic regime and to the Vlasov-Poisson 1D1V equation, which occur in plasma physics. The asymptotic-preserving properties of our procedure permit to study the long-time behavior of these models. In particular, we limit drastically by this method the numerical pollution, appearing in such time asymptotics when using classical numerical schemes.

show abstract

Section: Introductionmentioning

confidence: 99%

Asymptotic-Preserving Scheme for the Resolution of Evolution Equations with Stiff Transport Terms

Fedele¹,

Negulescu²,

Possanner³

2019

Multiscale Model. Simul.

View full text Add to dashboard Cite

show abstract

“…[11], [12]), and explicit-implicit flux splitting (see e.g. [13], [14], [15], [16]), constructed to allow for an efficient numerical solution of time-dependent problems. Another way to construct an all-speed algorithm is to develop an efficient way of solving the fully-implicit Euler system/NSEs.…”

Section: Introductionmentioning

confidence: 99%

An efficient algorithm for the multicomponent compressible Navier–Stokes equations in low- and high-Mach number regimes

Frolov

2019

Computers & Fluids

View full text Add to dashboard Cite

The goal of this study is to develop an efficient numerical algorithm applicable to a wide range of compressible multicomponent flows, including nearly incompressible low-Mach number flows, flows with strong shocks, multicomponent flows with high density ratios and interfacial physics, inviscid and viscous flows, as well as flows featuring combinations of these phenomena and various interactions between them. Although many highly efficient algorithms have been proposed for simulating each type of the flows mentioned above, the construction of a universal solver is known to be challenging. Extreme cases, such as incompressible and highly compressible flows, or inviscid and highly viscous flows, require different numerical treatments in order to maintain the efficiency, stability, and accuracy of the method.Linearized block implicit (LBI) factored schemes (see e.g.[1], [2]) are known to provide an efficient way of solving the compressible Navier-Stokes equations (compressible NSEs) implicitly, allowing us to avoid stability restrictions at low Mach number and high viscosity. However, the methods' splitting error has been shown to grow and dominate physical fluxes as the Mach number approaches zero (see [3]). In this paper, a splitting error reduction technique is proposed to solve the issue. A shock-capturing algorithm from [4] is reformulated in terms of finite differences, extended to the stiffened gas equation of state (SG EOS) and combined with the LBI factored scheme to stabilize the method around flow discontinuities at high Mach numbers. A novel stabilization term is proposed for low-Mach number applications. The resulting algorithm is shown to be efficient in both low-and high-Mach number regimes. Next, the algorithm is extended to a multicomponent case using an interface capturing strategy (see e.g. [5]) with surface tension as a continuous surface force (see [6]). Special care is taken to avoid spurious oscillations of pressure and generation of artificial acoustic waves in the numerical mixture layer. Numerical tests are presented to verify the performance and stability properties for a wide range of flows.

show abstract

“…The analysis [40,41,56,2,44,1] and the development of numerical methods [37,33,54,62,42,13,31,51,47,34,30,50,48,16,19,17,15,32,10,28,39,20,11,21] for the passage from compressible to incompressible gas dynamics has been and is still a very active field of research. The compressible Euler equations which describe conservation of density, momentum and energy in a fluid flow become stiff when the Mach number tends to zero.…”

Section: Introductionmentioning

confidence: 99%

“…However, this introduces strong drawbacks since the Mach number may become extremely small causing severe time step limitations. To circumvent these problems, in the recent past, asymptotic stable techniques have been developed [18,16,17,15,32,61,10,49,11,21]. These techniques permit to compute the solution of such stiff problems avoiding time step limitations directly related to the low Mach number regime.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, these methods lead to consistent approximations of the limit incompressible system when the Mach number goes to zero. In this context, in a recent work [21], a first order asymptotic preserving method has been developed. In particular, this work dealt with an analysis of the stability properties which led to a stability restriction on the numerical method independent from the Mach number.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

Dimarco

Loubère

Michel-Dansac

et al. 2018

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

In this work, we consider the development of implicit explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The scheme proposed is asymptotically stable with a CFL condition independent from the Mach number and it degenerates in the low Mach number regime to a consistent discretization of the incompressible system. Since, it has been proved that implicit schemes of order higher than one cannot be TVD (SSP) [29], we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing TVD methods in space. For this particular class of schemes, the TVD property is first proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first to the second order both in space and time, which preserves the monotonicity of the solution, highly accurate for all choices of the Mach number and with a time step only restricted by the non stiff part of the system. In the last part, we show thanks to one and two dimensional test cases that the method indeed possesses the claimed properties.

show abstract

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

Cited by 55 publications

References 36 publications

Asymptotic-Preserving Scheme for the Resolution of Evolution Equations with Stiff Transport Terms

Asymptotic-Preserving Scheme for the Resolution of Evolution Equations with Stiff Transport Terms

An efficient algorithm for the multicomponent compressible Navier–Stokes equations in low- and high-Mach number regimes

Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

Contact Info

Product

Resources

About