Stability of a Crank–Nicolson pressure correction scheme based on staggered discretizations

Boyer, Franck; Dardalhon, Fanny; Lapuerta, Céline; Latché, Jean-Claude

doi:10.1002/fld.3837

Cited by 5 publications

(13 citation statements)

References 26 publications

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“…The computed maximum drag and lift coefficients and the Strouhal number are c D = 3.24, c L = 1.01 and St = 0.300; these values belong to the reference intervals determined from the benchmark exercise [26], namely (3.22, 3.24), (0.99, 1.01) and (0.295, 0.305) respectively. They are also close to the values obtained with an algorithm based on the same space discretization but solving the equations of the asymptotic model for low Mach number flows [4].…”

Section: A Low Mach Number Casesupporting

confidence: 81%

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

Kheriji¹,

Herbin²,

Latché³

2013

Computers & Fluids

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SummaryWe assess in this paper the capability of a pressure correction scheme to compute shock solutions of the homogeneous model for barotropic two-phase flows. This scheme is designed to inherit the stability properties of the continuous problem: the unknowns (in particular the density and the dispersed phase mass fraction y) are kept within their physical bounds, and the entropy of the system is conserved, thus providing an unconditional stability property. In addition, the scheme keeps the velocity and pressure constant through contact discontinuities. These properties are obtained by coupling the mass balance and the transport equation for y in an original pressure correction step. The space discretization is staggered; the numerical schemes which are considered are the Marker-And Cell (MAC) finite volume scheme and the nonconforming low-order Rannacher-Turek and Crouzeix-Raviart finite element approximation. In either case, a finite volume technique is used for all convection terms. Numerical experiments performed here show that, provided that a sufficient dissipation is introduced in the scheme, it converges to the (weak) solution of the continuous hyperbolic system. Observed orders of convergence for 1D Riemann problems as a function of the mesh and time step at constant CFL number vary with the studied case, and the CFL number and on the regularity of the solution. They range from 0.5 to greater than 1 for the velocity and the pressure; in most cases, the density and mass fraction converge with a 0.5 order. Finally, the scheme shows a satisfactory behaviour up to large CFL numbers.

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Section: A Low Mach Number Casesupporting

confidence: 81%

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

Kheriji¹,

Herbin²,

Latché³

2013

Computers & Fluids

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“…Second, the non-dissipation of the kinetic energy is a prerequisite for numerical schemes for Large Eddy Simulation (e.g. [32,8,31,3]), and a theoretical proof of this feature thus strongly supports this kind of application. Finally, for Euler's equations, having at hand a discrete kinetic energy balance has been a key point in [24,18] to derive a consistent staggered scheme preserving the convex set of admissible states.…”

Section: Introductionmentioning

confidence: 94%

“…preserving the kinetic energy balance [25,1,3]. These techniques, implemented in the open-source software ISIS [27], have brought many outcomes, both from the theoretical and the practical points of view.…”

Section: Introductionmentioning

confidence: 99%

“…These techniques, implemented in the open-source software ISIS [27], have brought many outcomes, both from the theoretical and the practical points of view. First, the kinetic energy conservation property yields stability estimates (see [1,3] for quasi-incompressible flow and [13,18] for barotropic and non-barotropic compressible Navier-Stokes equations), and has been observed in numerical experiments to actually dramatically increase the reliability of the scheme. Second, the non-dissipation of the kinetic energy is a prerequisite for numerical schemes for Large Eddy Simulation (e.g.…”

Section: Introductionmentioning

confidence: 99%

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A convergent staggered scheme for the variable density incompressible Navier-Stokes equations

Latché¹,

Saleh

2017

Math. Comp.

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In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem (L ∞ -estimate for the density, L ∞ (L 2 )-and L 2 (H 1 )-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.Recently, for MAC, Crouzeix-Raviart and Rannacher-Turek approximations, discretizations of the convection operator in the momentum balance equation have been developed with the aim to obtain a scheme

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“…The computational domain is Ω = (−0.5, 0.5) 2 and the initial data consists in four constant states, in each of the four sub-squares of Ω obtained by splitting it along the lines joining the mid-points of each segment of the boundary (i.e. in Ω 1,1 = (−0.5, 0) × (0, 0.5), Ω 1,2 = (0, 0.5) 2 , Ω 2,1 = (−0.5, 0) 2 and Ω 2,2 = (0, 0.5) × (−0.5, 0)). These constant states are chosen in such a way that each associated onedimensional Riemann problem (i.e.…”

Section: A Two-dimensional Riemann Problemmentioning

confidence: 99%

An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations

Grapsas¹,

Herbin²,

Kheriji³

et al. 2016

The SMAI Journal of Computational Mathematics

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International audienceIn this paper we present a pressure correction scheme for the compressible Navier-Stokes equations. The space discretization is staggered, using either the Marker-And Cell (MAC) scheme for structured grids, or a nonconforming low-order finite element approximation for general quandrangular, hexahedral or simplicial meshes. For the energy balance equation, the scheme uses a discrete form of the conservation of the internal energy, which ensures that this latter variable remains positive; this relation includes a numerical corrective term, to allow the scheme to compute correct shock solution in the Euler limit. The scheme is shown to have at least one solution, and to preserve the stability properties of the continuous problem, irrespectively of the space and time steps. In addition, it naturally boils down to a usual projection scheme in the limit of vanishing Mach numbers. Numerical tests confirm its potentialities, both in the viscous incompressible and Euler limits

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Stability of a Crank–Nicolson pressure correction scheme based on staggered discretizations

Cited by 5 publications

References 26 publications

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

A convergent staggered scheme for the variable density incompressible Navier-Stokes equations

An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations

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