Over the last ten years, robustness of schemes has raised an increasing interest among the CFD community. One mathematical aspect of scheme robustness is the positivity preserving property. At high Mach numbers, solving the conservative Euler equations can lead to negative densities or internal energy. Some schemes such as the flux vector splitting (FVS) schemes are known to avoid this drawback. In this study, a general method is detailed to analyze the positivity of FVS schemes. As an application, three classical FVS schemes (Van Leer's, Hänel's variant and Steger and Warming's) are proved to be positively conservative under a CFL-like condition. Finally, it is proved that for any FVS scheme, there is an intrinsic incompatibility between the desirable property of positivity and the exact resolution of contact discontinuities

Gressier, Jérémie

Moschetta, Jean-Marc

English

Open Archive Toulouse Archive Ouverte

ON THE POSITIVITY OF FVS SCHEMESJ.-M. MOSCHETTA, J. GRESSIERDepartment of Aerodynamics, SUPAERODepartment Models for Aerodynamics and Energetics, ONERABP 4025, 31055 Toulouse CEDEX 4, FranceE-mail: moscheta@onecert.frAbstractOver the last ten years, robustness of schemes has raised an increasing inter-est among the CFD community. One mathematical aspect of scheme robust-ness is the positivity preserving property. At high Mach numbers, solvingthe conservative Euler equations can lead to negative densities or internalenergy. Some schemes such as the flux vector splitting (FVS) schemes areknown to avoid this drawback. In this study, a general method is detailedto analyze the positivity of FVS schemes. As an application, three classicalFVS schemes (Van Leer’s, Ha¨nel’s variant and Steger and Warming’s) areproved to be positively conservative under a CFL-like condition. Finally,it is proved that for any FVS scheme, there is an intrinsic incompatibil-ity between the desirable property of positivity and the exact resolution ofcontact discontinuities.1. IntroductionIn highly accelerated flows, the total energy is mainly composed of ki-netic energy. Yet, in conservative formulation, both total and kinetic en-ergy are computed independently, and their difference may yield negativeinternal energy, aborting the computation. In order to give some math-ematical interpretation of schemes robustness or weakness in such severeconfigurations, it is useful to introduce the positivity property: a schemeis said to be positively conservative if, starting from a set of physicallyadmissible states, it can only compute new states with positive densitiesand internal energies. Perthame (Perthame, 1990) first proposed a schemewhich satisfies this property. Afterwards, Einfeldt et al.(Einfeldt, 1991)2 J.-M. MOSCHETTA, J. GRESSIERgave some results concerning Godunov-type schemes. They proved thatGodunov scheme is positively conservative while Roe’s scheme is not, andderived the HLLE method, a positive variant of HLL schemes family. Later,Villedieu and Mazet (Villedieu, 1995) proved that Pullin’s EFM kineticscheme (Pullin, 1980) is positively conservative under a CFL-like condi-tion. Recently, Dubroca (Dubroca, 1998) proposed a positive variant ofRoe’s method. Since any scheme is positively conservative for a zero timestep, it is absolutely essential to specify a time step condition when definingthe positivity property.2. Defining scheme positivitySince one can formally extend any first-order one-dimensional positivelyconservative method to a second-order multidimensional positively conser-vative method (Perthame, 1996; Linde, 1997), we will restrict ourselves tothe case of first-order schemes for the one-dimensional Euler equations inthe following analysis. A conservative explicit method applied to the Eulerequations can be expressed asUi = Ui − ∆t∆x[Fi+1/2 − Fi−1/2](1)where Ui is the average value over cell Ωi of the vector of conservativevariables T (ρ, ρu, ρE) at a given time step. Ui is the updated state vector.∆x is the measure of cell Ωi and Fi+1/2 is the numerical flux between thecells Ωi and Ωi+1. The numerical flux is a function Fi+1/2 = F (Ui,Ui+1) ofthe states in both neighboring cells. The numerical flux must satisfy theconsistency condition F (U ,U) = F(U), where F is the exact Euler flux.The discretized conservation equation Eq. (1) can be rewritten asUi = Ui − χlociλi[Fi+1/2 − Fi−1/2](2)where λ(U) is the characteristic wave speed defined by λ(U) = |u|+ a andχloci = λ(Ui)∆t/∆x. For physical reasons, the state U cannot take anyarbitrary value in R3. Its density and internal energy must be both strictlypositive. One can define ΩU as the open set of physically admissible statesΩU ={U = T (u1, u2, u3) | u1 > 0 and 2u1u3 − u22 > 0} (3)Vacuum is an admissible state for the closure ΩU but not for ΩU since itis not expected to be reached in practical computations.POSITIVITY OF FVS SCHEMES 3Definition 1 A scheme is said to be positively conservative if and only ifthere exists a constant χ, such that if both conditions are satisfied• ∀i ∈ Z , Ui ∈ ΩU (4a)• ∆t ≤ χ ∆xmaxi∈Zλ(Ui) (4b)then∀i ∈ Z , Ui ∈ ΩU (5)For ∆t = 0, according to Eq. (1), one has ∀i ∈ Z , Ui = Ui ∈ ΩU for anyflux function used. So, for any continuous flux function F , since ΩU is anopen subset of R3, whatever initial conditions Ui are in ΩU , one can find ∆tsmall enough which will preserve positivity of states Ui. Consequently, theproperty of positivity consists of proving that ∆t is not too small comparedto the maximum time step given by the CFL condition. Otherwise, one canfind a situation in which a physical admissible state can only be obtainedby a vanishing time step, which is not acceptable for practical gas dynamicsapplications. On the contrary, a scheme is said to be non-positive if∀ χ > 0 , ∃ (U)i∈Z ∈ ΩU , Ui /∈ ΩU (6)For a non-positive scheme (e.g. Roe, AUSM), one may have to use anextremely small time step to update the solution and may not be able toproduce a physically admissible solution after a finite period of time.3. Positivity of FVS methodsThis study has been restricted to a class of FVS schemes in which the fluxesF± satisfy the symmetry propertyF−(U) = −F+(U) (7)where X is the symmetric vector T (x1,−x2, x3) of X = T (x1, x2, x3). Itshould also satisfy∀u, a ∈ R× R+ , limρ→0F±(ρ, u, a) = 0 (8)Since F±(U) is generally an homogeneous function of ρ, Eq. (8) is not arestrictive assumption in practice.4 J.-M. MOSCHETTA, J. GRESSIERTheorem 1 A given consistent FVS scheme satisfying properties (7) and(8) is positively conservative if and only if its F± functions satisfy bothproperties:• ∀U ∈ ΩU , F+(U) ∈ ΩU (9a)• ∃χ > 0 , ∀U ∈ ΩU , U − χλ(U) [F+(U)− F−(U)] ∈ ΩU (9b)In that case, the less restrictive positivity condition is expressed as∀i ∈ Z , χloci < χopt (10)where χopt is the greatest constant χ satisfying (9b).Proof A detailed proof can be found in (Gressier, 1999).The condition (9b) leads to a maximum time step which has then to be putinto a CFL-like form χloc < χopt. This is the case for VL, VLH and SWschemes since χopt = infM (χmax) is not zero. It turns out that the inter-VL VLH SWF+Supersonic |M | ≥rγ − 12γSubsonic γ ≥ 1 1 ≤ γ ≤ 3W′Supersonic χloc <|M |+ 1|M |+qγ−12γSubsonic χloc < χV Lmax χloc< χV LHmax χloc< χSWmaxTABLE 1. Internal energy positivity conditionsnal energy positivity conditions are always more stringent than the masspositivity conditions. Therefore, it is the internal energy positivity condi-tion which actually drives the scheme positivity. Moreover, it means thatzero values cannot be reached simultaneously by density and internal en-ergy. Since expressions of χV Lmax, χV LHmax and χSWmax are intricate, they are notdetailed but these coefficients can be easily computed as a function of thelocal Mach number. The smallest values of these conditions have been com-puted and lead to the optimal CFL condition χopt which ensures that thescheme is positively conservative in all configurations. These constants χoptare summarized in table 2 and lead to an optimal CFL number of one forPOSITIVITY OF FVS SCHEMES 5VL VLH SW1 min„1,2γ«1TABLE 2. Optimal CFL numberχopt.0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0M0.00.20.40.60.81.01.21.41.61.8van Leer  van Leer/HänelSteger-WarmingFully UpwindFigure 1. Maximum CFL number χloc to ensure internal energy positivity.usual gases where 1 < γ < 2. Since necessary and sufficient conditions havebeen derived, it can be interesting to plot the local CFL conditions. Thedifferent χmax functions are plotted on Fig. 1 for γ = 1.4. The SW schemeyields the most severe condition while the VL scheme allows a greater localCFL condition in the subsonic range. All three curves merge in the super-sonic range where the CFL condition implies that χ should decrease to 1for high Mach numbers (Fig. 1). As a consequence, a CFL number of one afortiori ensures positivity of the three schemes. According to Fig. 1, a CFLnumber of 1.45 (for γ = 1.4) can be used with Van Leer’s method if theflow is expected to remain subsonic. Note that this condition only ensuresthe scheme positivity, but not its stability. Using too high CFL numbersmight produce oscillations even though the updated solution would still bean admissible state.6 J.-M. MOSCHETTA, J. GRESSIER4. Accuracy versus PositivityMost FVS schemes have proved to be robust in many flow configurationsbut none of them are able to exactly resolve contact discontinuities sinceit remains a non-vanishing dissipation which smears out an initial discon-tinuity of density. Van Leer (Van Leer, 1991) pointed out that preventingnumerical diffusion of contact discontinuities may lead to a marginally sta-ble or unstable behavior for slow flows.Theorem 2 If a FVS scheme exactly preserves stationary contact discon-tinuities, then it cannot be positively conservative.Proof Consider a FVS scheme given by its flux functions F±, and assumeit exactly preserves stationary contact discontinuities. Then, the interfaceflux between UL = T(ρL, 0,pγ−1)and UR = T(ρR, 0,pγ−1)must satisfyF+(UL) + F−(UR) = (0, p, 0)T (11)Since ρL and ρR are independent variables, F+(UL) must be a function ofonly p. Hence, for all U = T(ρ, 0, pγ−1),F+(U) = (f1(p), f2(p), f3(p))T (12a)Moreover, considering the symmetry property (7) and using U = U , onehas F−(U) = −F+(U). Then,F−(U) = (−f1(p), +f2(p), −f3(p))T (12b)Substituting expressions (12a) and (12b) in Eq. (11), one obtains f2(p) =p/2. Moreover, f1(p) must be positive or null to satisfy the condition (9a) ofpositivity. If f1(p) = 0, condition (9a) is not satisfied since f2(p) is not null.If f1(p) > 0, then the first component of W′ = U − χlocλ[F+(U) − F−(U)]may be expressed asρ− χloca2f1(p) = ρ−√ρ(2χlocf1(p)√γp)(13)Hence, for all functions f1(p) and for all χloc > 0, one can always find pand ρ such that expression (13) is negative.5. ConclusionA general method to prove the positivity of FVS schemes has been proposedand applied to standard FVS schemes, namely the van Leer scheme, one ofPOSITIVITY OF FVS SCHEMES 7its variants, and the Steger and Warming scheme. Although these schemeshave been known for a long time to be robust, they are now proved tobe positively conservative under a CFL condition of 1, for usual valuesof the specific heat ratio γ in the range [1; 2]. In particular, this showsthat all these FVS schemes can be confidently applied to gas dynamicsproblems including real gas effects for which γ may range between 1.4 and1. Moreover, these conditions have been proved to be incompatible with theparticular form of FVS schemes which would be able to exactly preservestationary contact discontinuities. In other words, one cannot develop arobust and accurate scheme within the FVS family.ReferencesDubroca B (1998). Positively Conservative Roe’s Matrix for Euler Equations. LectureNotes in Physics, pp 272–277. 16th ICNMFD, Springer Verlag, 1998.Einfeldt B, Munz C D, Roe P L, Sjo¨green B (1991). On Godunov-Type Methods nearLow Densities J. Comput. Phys. 92, pp 273-295.Gressier J, Villedieu P, Moschetta J M (1999). Positivity of Flux-Vector SplittingSchemes. J. Comput. Phys. 155, pp 199–220.Linde T, Roe P L (1997). Robust Euler Codes. AIAA Paper 97-0209.Perthame B (1990). Boltzmann type Schemes for Gas Dynamics and the Entropy Prop-erty. SIAM J. Numer. Anal. 27, pp 1405-1421.Perthame B, Shu C (1996). On Positive Preserving Finite Volume Schemes for the Com-pressible Euler Equations Numer. Math. 76, pp 119-130.Pullin D I (1980). Direct Simulation Methods for Compressible Inviscid Ideal-Gas Flow.J. Comput. Phys. 34, pp 231-244.Van Leer B (1991). Flux-Vector Splitting for the 1990s NASA CP-3078.Villedieu P, Mazet P (1995). Kinetic Schemes for the Euler Equations Out of Thermo-chemical Equilibrium. La Recherche Aerospatiale 2, pp 85-102.

Positively Conservative Roe’s Matrix for Euler Equations.

On the positivity of fvs schemes

http://oatao.univ-toulouse.fr/2394/1/Gressier_2394.pdf

On the positivity of fvs schemes

Abstract

Similar works

Full text

Available Versions

Open Archive Toulouse Archive Ouverte