Positivity of Flux Vector Splitting Schemes

Gressier, Jérémie; Villedieu, Philippe; Moschetta, Jean‐Marc

doi:10.1006/jcph.1999.6337

Cited by 60 publications

(39 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It is known that many first order Godunov-type schemes [2,16,3] have the so called positivity-preserving property and can maintain positive density and pressure. It has been also proved that some second-order conservative schemes [15,5] are positivity-preserving with or without a more restrictive CourantFriedrichs-Lewy (CFL) condition.…”

Section: Introductionmentioning

confidence: 99%

Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

Adams

Shu

2013

Journal of Computational Physics

188

140

View full text Add to dashboard Cite

In this work a simple method to enforce the positivity-preserving property for general high-order conservative schemes is proposed. The method keeps the original scheme unchanged and detects critical numerical fluxes which may lead to negative density and pressure, and then imposes a cut-off flux limiter to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. A number of numerical examples suggest that this method, when applied on an essentially non-oscillatory base scheme, can be used to prevent positivity failure when the flow involves vacuum or near vacuum and very strong discontinuities.

show abstract

Section: Introductionmentioning

confidence: 99%

Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

Adams

Shu

2013

Journal of Computational Physics

188

140

View full text Add to dashboard Cite

show abstract

“…al. [6] gave a general positivity analysis for the FVS schemes, which include the van Leer and the Steger-Warming's FVS methods. Estivalezes and Villedieu [4] constructed a general framework to transform a positive FVS scheme into a positive multidimensional higher-order accurate scheme with the implementation of anti-diffusive terms.…”

Section: Introductionmentioning

confidence: 99%

Pseudoparticle representation and positivity analysis of explicit and implicit Steger-Warming FVS schemes

Tang¹,

Xu²

2001

Z. angew. Math. Phys.

View full text Add to dashboard Cite

This paper is about the pseudo-particle representation and the positivity analysis of an explicit and an implicit Steger and Warming's flux vector splitting (FVS) scheme for the compressible Euler equations. The positivity proof is based on the motion of pseudo-particles. For the explicit scheme, it shows that the density and the internal energy could keep non-negative values under the CFL-like condition for the Steger-Warming FVS scheme once the initial gas stays in a physically realizable state. For the implicit method, under a stronger CFL-like condition, the positivity property can also be preserved. Mathematics Subject Classification (2000). 35L65, 65M06, 76M20, 76K05.

show abstract

“…More general relaxation approximations for systems were proposed in [31]. These enable to avoid the problem of excessive diffusion on contact discontinuities that are inherent to kinetic methods [48,44]. Relaxation approximations that well resolve contact discontinuities have been then developed, making the link to approximate Riemann solvers [18,17,21].…”

Section: The Kinetic Bgk Methodsmentioning

confidence: 99%

Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations

Bouchut¹,

Jobic²,

Natalini³

et al. 2018

The SMAI Journal of Computational Mathematics

View full text Add to dashboard Cite

Abstract. Kinetic BGK numerical schemes for the approximation of incompressible Navier-Stokes equations are derived via classical discrete velocity vector BGK approximations, but applied to an inviscid compressible gas dynamics system with small Mach number parameter, according to the approach of Carfora and Natalini (2008). As the Mach number, the grid size and the timestep tend to zero, the low Mach number limit and the time-space convergence of the scheme are achieved simultaneously, and the numerical viscosity tends to the physical viscosity of the Navier-Stokes system. The method is analyzed and formulated as an explicit finite volume/difference flux vector splitting (FVS) scheme over a Cartesian mesh. It is close in spirit to lattice Boltzmann schemes, but it has several advantages. The first is that the scheme is expressed only in terms of momentum and mass compressible variables. It is therefore very easy to implement, and several types of boundary conditions are straightforward to apply. The second advantage is that the scheme satisfies a discrete entropy inequality, under a CFL condition of parabolic type and a subcharacteristic stability condition involving a cell Reynolds number that ensures that diffusion dominates advection at the level of the grid size. This ensures the robustness of the method, with explicit uniform bounds on the approximate solution. Moreover the scheme is proved to be second-order accurate in space if the parameters are well chosen, this is the case in particular for the Lax-Friedrichs scheme with Mach number proportional to the grid size. The scheme falls then into the class of artificial compressibility methods, the novelty being its exceptionally good theoretical properties. We show the efficiency of the method in terms of accuracy and robustness on a variety of classical two-dimensional benchmark tests. The method is finally applied in three dimensions to compute the permeability of a porous medium defined by a complex idealized Kelvin-like cell. Relations between our scheme and compressible low Mach number schemes are discussed.

show abstract

Positivity of Flux Vector Splitting Schemes

Cited by 60 publications

References 19 publications

Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

Pseudoparticle representation and positivity analysis of explicit and implicit Steger-Warming FVS schemes

Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations

Contact Info

Product

Resources

About