Kinetic derivation of a finite difference scheme for the incompressible Navier Stokes equation

Transcription

1 deriation of a finite difference scheme for the incompressible Naier Stokes eqation Mapndi K. Banda Michael Jnk Axel Klar Abstract In the present paper the low Mach nmber limit of kinetic eqations is sed to deelop a discretization for the incompressible Naier-Stokes eqation. The kinetic eqation is discretized with a first and second order discretization in space. The discretized eqation is then considered in the low Mach nmber limit. Using this limit a second order discretization for the conectie part in the incompressible Naier Stokes eqation is obtained. Nmerical experiments are shown comparing different approaches. Kewords. kinetic eqations, asmptotic analsis, low Mach nmber limit, second order pwind discretization, slope limiter, incompressible Naier-Stokes eqation AMS sbject classifications. Introdction eqations or discrete elocit models of kinetic eqations ield in the limit for small Kndsen or Mach nmbers an approximation of macroscopic eqations like the Eler or incompressible Naier Stokes eqation. Discretizations of kinetic models are often sed in combination with the aboe limiting procedres to deelop discretizations for the corresponding macroscopic limit eqation. We consider first the Eler or hdrodnamic limit: As a first example of the aboe approach we mention the Schemes. A simple kinetic relaxation model with a so called BGK operator is discretized. In the limit a scheme for the hperbolic limit eqation is obtained. There is a large amont of literatre on the sbject, examples can be fond in [, 0, 6, 7, 5]. A second example is gien b the relaxation schemes deeloped in [4]. Based on a simplified kinetic model efficient second order pwind discretizations for the Eler eqation are deeloped. For frther examples and related schemes, see [3, 9,, 8]. This work was spported b DAAD grant and b DFG grant Department of Mathematics, Technical Uniersit of Darmstadt, 6489 Darmstadt, German. Department of Mathematics, Uniersit of Kaiserslatern, Kaiserslatern, German.

2 As alread mentioned, another scaling - the diffsie scaling - gies in the limit the incompressible Naier Stokes eqation. Again kinetic models - sall based on a small nmber of elocities - are discretized and inestigated in the incompressible Naier Stokes limit. A well known example are the Lattice-Boltzmann methods, see [9, 4,, 8]. Also relaxation schemes for diffsie limits hae been deeloped for example in [3, 7]. See also [8, ] for related schemes for the limit eqations. In the present paper we start b recalling the diffsie scaling of kinetic eqations, leading to the incompressible Naier Stokes eqation. Then, a natral discretization of the kinetic eqation is sed to obtain in the limit a second order slope limiting procedre for the conectie term of the Naier Stokes eqation. For the slope limiting procedre the behaior of the soltion in all spatial directions is important not onl along the ales of the soltion along the coordinate axis. The paper is organized as follows: Section contains a short description of the reslts of the asmptotic analsis leading from kinetic eqations to the incompressible Naier Stokes eqation. In section 3 the asmptotic procedre is performed for the discretized kinetic eqations and a general limit discretization for the incompressible Naier Stokes eqation is deried. In 4 we concentrate on the deriation of the discretization of the conectie part. A first and second order pwind discretization for the limit eqation is deried. Whereas the first order discretization is standard, the second order discretization incldes a mltidimensional slope limiting procedre. Eqations and the Incompressible Naier Stokes Eqation The incompressible Naier Stokes eqation t + + x p = µ x, di x = 0. is a reasonable model to describe the large scale and long time behaior of a slowl flowing isothermal gas. We will se the fact that. can be formall obtained as scaling limit of a Boltzmann tpe kinetic eqation t f + x f = Jf.. Here, f = fx,, t is the phase space densit of the gas atoms which we consider, for simplicit, in the two dimensional case x = x, x R, =, R. We will not specif the complete strctre of the collision operator Jf. Onl those properties which are important in the Naier Stokes limit will be listed below. Using the diffsie space-time scaling x x/ɛ, and t t/ɛ, the restriction to large scale and long time behaior is incorporated. We find the scaled kinetic eqation t f + ɛ xf = Jf..3 ɛ

3 The assmption of er slow, isothermal flows which exhibit onl small densit ariations is then taken care of b assming that f is onl a small pertrbation of the Maxwellian elocit distribtion M M = π exp, R which corresponds to the constant state: densit one, elocit zero, and temperatre one. Or precise assmption on the strctre of f is f = M + ɛg, g = g 0 + ɛg + ɛ g The scaled eqation.3 together with.4 is the standard pertrbation procedre to obtain. as limiting problem see [, 5, ]. In the next section, we demonstrate this procedre in a slightl more general sitation where.3 is modified b adding a diffsie term D h f and replacing x with an approximation h x. Let s now list some properties of J which will be needed for the analsis: if.4 is inserted into.3 we need a Talor expansion of JM + ɛmg to compare terms of eqal order in ɛ. We hae M JM + ɛmg = ɛlg + ɛ Qg, g + ɛ 3 Rg.5 where L inoles the first and Q the second Frechet deriatie of J at the point M see [] for details. The strctre of the remainder R is not releant in the limit. Note that the zero order term in.5 drops ot becase of the assmed eqilibrim condition JM = 0..6 Another important assmption is that the collision inariants of J are the fnctions,, in isothermal flows is not a collision inariant, which means in terms of the weighted L scalar prodct g, h = R ghm d M Jf, ψ = 0, ψ {,, }..7 Note that.7 implies together with.5 that also Lg, ψ = Qg, g, ψ = Rg, ψ = 0.8 for all collision inariants ψ. Important assmptions on the operator L are L is selfadjoint with respect to, and Lh, h 0. L satisfies a Fredholm alternatie with a three dimensional kernel spanned b the collision inariants we denote the psedo inerse of L b L, i.e. L L is the orthogonal projection onto kerl. 3

4 Additionall, with µ, λ > 0 L k l, i = 0, L k l, i j = λδ kl δ ij µδ ik δ jl + δ il δ jk,.9 which wold be a conseqence of inariance of J nder certain coordinate transformations in the elocit space as well as L being self adjoint and semi definite. Finall, we need a propert of Q which is a direct conseqence of the relation Qh, h = Lh for h = α + β.0 see [] for the deriation. Using the fact that,, are in the kernel of L and that L L is the projection onto kerl, we conclde L Qh, h = β i β j L L i j = β i β j i j δ ij.. The projection has been calclated with the sal Schmidt procedre which reqires the scalar prodcts, i j and k, i j. Sch moments of the standard Maxwellian will be freqentl sed later and we list them here for conenience, =,, i = 0, i, j = δ ij i j, k =, i j, k l = δ ij δ kl + δ ik δ jl + δ jk δ il.. 3 The discretized kinetic eqation and deriation of macroscopic discretization We start with the kinetic eqation. which is discretized sing the method of lines t f + h x f D hf = Jf. In this section, we onl assme that h x = x h, x h and D h are linear operators, that h x is independent of, and that the components of h x commte. In the next section we choose x h i as central difference approximations and D h as nmerical iscosit term. Introdcing the diffsie scaling as in the preios section and rescaling D h f according to ɛ D h f, we find t f + ε h x f D hf = ε Jf. With the expansion.4 and.5, we then get t g + ε h x g D hg = ε Lg + Qg, g + Rg. ε and the reglar expansion g = g 0 + εg + ε g +... leads to t g + ε h x g D hg = ε Lg 0 + Lg + ε Qg 0, g 0 + Lg + Qg 0, g + Rg 0 + Oε 3. 4

5 In lowest order ɛ we immediatel find Lg 0 = 0 so that g 0 kerl, i.e. g 0 = ρ +, ρ R, R 3. with parameters ρ, which are et ndetermined. In order ɛ, we hae h xg 0 = Lg + Qg 0, g and with.8 we conclde h xg 0, = 0, h xg 0, i = Using 3., the first condition can be reformlated 0 = h x i g 0, i = h x i ρ + j j, i =, i h x i ρ + j, i h x i j = h x i i = : di h x where we hae sed moment relations from.. Similarl, we find so that 3.4 implies 0 = h x j j g 0, i = j, i h x i ρ + j k, i h x i k = h x i ρ di h x = 0, h xρ = which has to be satisfied b the parameters ρ, in 3.. Appling L to 3.3, we obtain the projection of g onto kerl, so that with additional parameters ρ, g = L h xg 0 Qg 0, g 0 + ρ +. Using 3.5, we can calclate h xg 0 = i h x i ρ + j j = i j h x i j and hence L h xg 0 = h x i j L i j. The expression L Qg 0, g 0 can be simplified with., so that g = h x i j L i j + i j i j δ ij + ρ + j j. 3.6 Going back to 3. and collecting terms of order ɛ 0, we find t g 0 + h xg D h g 0 = Lg + Qg 0, g + Rg Again, propert.8 ields conditions on the ndetermined parameters in g 0 and g. Integrating 3.7 and obsering that g, i = i becase of.9 and., we get a diergence condition on t ρ + di h x = D h g 0,. Integration of h xg after mltiplication with ψ = j ields with the help of.9 and. h x i i g, j = h x i h x k l λδ kl δ ij µδ ik δ jl + δ il δ jk + h x i l k δ ik δ jl + δ il δ jk + h x j ρ 5

6 so that the j weighted integral of 3.7 leads to t j + h x i i j D h g 0, j + h x j ρ = λ h x j h x k k + µ h x k h x j k + µ h x i h x i j. Taking into accont that h x and h x commte, relation 3.5 implies that λ h x j h x k k = µ h x k h x j k = 0. Introdcing the discretized Laplacian h x = h x i h x i, we hae the final reslt t j + x h i i j D h g 0, j + x h j ρ = µ h x j, di h x = Note that 3.8 redces to the incompressible Naier Stokes eqation. if we choose h x = x and D h = 0. Obiosl, ρ takes the role of the pressre and h x i i j D h g 0, j gies a discretization of the conectie terms ones the nmerical iscosit is fixed. 4 First and second order pwind schemes To find expressions for h x and the nmerical iscosit D h we consider the linear transport part of the kinetic eqation in two dimensions: A first order discretization is gien b with positie constants c, c and x f = x f + x f. 4. x h f + x h f c h,h x f c h,h x f 4. h x f ij = h f i+j f i j,,h x f ij = h f i+j f ij + f i j, h x f ij = h f ij+ f ij,,h x f ij = h f ij+ f ij + f ij. The constants are chosen later according to the macroscopic flow sitation nder consideration. In iew of 4., we define h h D h f = c,h x f + c,h x f and obtain D h g 0 = c x,h ρ h + c x,h ρ h + c x,h h i i + c x,h h i i which ields with. the reqired expressions in 3.8 D h g 0, = h c x,h + c x,h. 6

7 S PSfrag replacements S x 0 S 0 0 x x Figre : Piecewise linear definition of φx, in the sets S 0, S, S A second order discretization of 4. is obtained b slope limiting [ h x f c i, j ij ϕ h i+ j i+ jf ϕ i j i jf + c i, j ] 4.3 ϕ h ij+ ij+ f ϕ ij ij f where h x are again central differences, the f increments are defined b and i+ jf = f i+j f ij, ij+ f = f ij+ f ij, ϕ i+ ϕ ij+ j = ϕr i+ j, r i+ j = i jf/ i+ jf = ϕr ij+, r ij+ = ij f/ ij+ f, with ϕr = max{0, min{r, }} being the minmod limiter. Using the definition of ϕ, one can write expressions like ϕ i+ j i+ jf as φ i jf, i+ jf where φ is a continos, piecewise linear fnction on R defined according to figre. Extracting the iscosit term in 4.3, we get D h f ij = c i, j φ h i jf, i+ jf φ i 3 jf, i jf + c i, j ] φ h ij f, ij+ f φ ij 3 f, ij f Now, the task to calclate D h g 0, j is more inoled than in the first order case. We start with the obseration that i+ jg 0 = i+ j becase ρ satisfies h x ρ = 0. Hence, a tpical term appearing in D hg 0, has the form φδ, δ, δ, δ R

8 Introdcing the linear map δ δ T = δ δ we can rewrite 4.4 as φt,. Note that φ is linear in the conex sets S 0, S, S see figre. With the nit ectors e =, 0, e = 0, and Sa, b = S + a, b S a, b, S ± = {±λ a + λ b : λ, λ 0}, we can describe these sets as S 0 = Se, e + e, S = e + e, e, S = Se, e. Assming that T is inertible, we conclde that φ T is linear on the sets Ŝ i = T S i. Since Sa, b = csa, b for all c 0, we hae Ŝi = ˆT S i where ˆT = det T T δ δ = = δ δ δ δ. Hence, Ŝ 0 = S δ, δ δ, Ŝ = Sδ δ, δ, Ŝ = Sδ, δ. Taking into accont that φ anishes on S 0, we find φt, = δ δ M d + δ M d Ŝ Ŝ or with the help of the matrix aled fnction Ia, b = M d that Sa,b φt, = Iδ δ, δ δ δ + Iδ, δ δ = : Lδ, δ. Hence, the nmerical iscosit for the second order discretization has the form D h g 0, ij = c i, j L h i j, i+ j L i 3 j, i j + c i, j ] L h ij, ij+ L ij 3, ij We conclde b giing an explicit formla for the fnction I. First, we note that, sing the smmetr of M and Ia, b = M d. S + a,b Using that S + a, b is a cone with some opening angle 0 < β < π arond the 8

9 PSfrag replacements α b S + a, b β a Figre : Angles α, β characterizing the cone S + a, b ra in direction α see figre, we go oer to polar coordinates and find α+β/ cos Ia, b = ψ sin ψ cos ψ r α β/ sin ψ cos ψ sin dψ ψ 0 π e r r dr. After some straight forward calclations we get Ia, b = cosα sinα δ + sin δ. π sinα cosα In the case where T is not inertible, one can either slightl modif the row ectors δ, δ in order to obtain an inertible mapping note that φt, is continos in T, or one can se the relation φt, j = φt e j. 4.5 To proe 4.5, let s consider the case δ = γδ. Then, T = δ γ and φt = δ φ, γ. Now, δ, j = δ e j so that 4.5 follows. The case δ = γδ can be treated similarl. 5 Nmerical reslts We inestigate the aboe first and second order discretizations of the conectie term nmericall for the following stationar eqation and compare them with the second order scheme described in [8]. The iscos term ε is treated b straightforward second order central differences. x ε = 0 + bondar conditions To test or scheme for pre conection we se test examples for the stationar case as presented in [0]. The first example is pre conection of a Step Profile, i.e. the following flow sitation is considered: Consider x, x in [0, ]. The domain of comptation is diided into two sbdomains which gie a step profile as sketched below. 9

10 PSfrag replacements PSfrag replacements 0, 0, tan θ/, 3D Illstration of Step Profile 0,0,,0 x 0,0 = θ tan θ/ x = = cos θ sin θ x x, x x x = = tan θ/ cos θ = 0.6 sin θ 0.4 x θ Bondar ales are chosen according to the respectie domain. The soltion domain is discretized sing a 4 4 reglar mesh for different flow angles θ. We ma choose c and c constant proportional to the maximal flow elocit: c = max ij { ij }, c = max ij { ij }. Alternatiel the local flow elocit can be sed: c i, j = max{ i+j, ij }, c i, j = max{ ij+, ij } In or nmerical tests the local flow elocit is sed. The reslting nonlinear sstem is soled b a GMRES-based soler described and implemented b Kelle [6]. The reslts are plotted in the following figres. The show the compted profile at the line x = for both the elocit components and. In the figres we make a comparison of the first order pwind method, the second order approach b Krgano and Tadmor [8] and the kinetic second order approach deeloped here. We alwas se the local flow elocit to determine c and c..6.5 Pre conection of a Step Profile, θ = π/4,ε = Pre conection of a Step Profile, θ = π/4,ε =

11 .7.6 Pre conection of a Step Profile, θ = 35,ε = 0.. Pre conection of a Step Profile, θ = 35,ε = Pre conection of a Step Profile, θ = 5,ε = 0 Pre conection of a Step Profile, θ = 5,ε = The second example is pre conection of a box-shaped profile: We consider a box-shaped profile as shown in the figre below. As in the preios example we compare approximations of the profile across a ertical plane in the middle PSfrag of the replacements soltion domain. We compare PSfrag the different replacements schemes sing a niform 4 4 mesh. 0, 0,, 3D Illstration of Box Profile 0,0 = x 45 0,0 = x = x x, x x 0.6,,0 x x = = 45

12 .6.5 Pre conection of a Box Profile, θ = π/4,ε = 0,* = Pre conection of a Box Profile, θ = π/4,ε = 0, * = Pre conection of a Box Profile, θ = π/4,ε = 0, * = Pre conection of a Box Profile, θ = π/4,ε = 0, * = The figres show that in the cases considered here the reslts are comparable to those obtained b the Krgano-Tadmor approach. Frther, the conection-diffsion eqation for the step-profile is considered. We choose ε = Jst like in the preios examples we took a niform 4 4 mesh..6.5 Conection Diffsion of a Step Profile, θ = π/4,ε = Conection Diffsion of a Step Profile, θ = π/4,ε =

13 .6.5 Conection Diffsion of a Step Profile, θ = π/4 ε = 0 ε = Conection Diffsion of a Step Profile, θ = π/4 ε = 0 ε = Conection Diffsion of a Box Profile, θ = π/4 ε = 0 ε = Conection Diffsion of a Box Profile, θ = π/4 ε = 0 ε = Conclsion Starting from special discretizations of the Boltzmann eqation we hae obtained corresponding discretizations of the incompressible Naier Stokes eqation in the diffsie, low Mach nmber limit. In particlar, a second order slope limiting approach on the kinetic leel gies rise to a limiter for the nonlinear term in the Naier Stokes eqation which exhibits an intersting strctre. From the simple nmerical tests one can conclde that the new scheme ields reslts which are comparable to those obtained b other second order methods. A test of the new limiter in more complicated flow sitations with strongl locall aring elocit fields is in preparation. References [] Krgano A. and D. Le. A third-order semi-discrete central scheme for conseration laws and conection diffsion eqations. preprint. [] C. Bardos, F. Golse, and D. Leermore. Flid dnamic limits of kinetic eqations: Formal deriations. J. Stat. Phs., 63:33 344, 99. 3

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15 [8] A. Krgano and E. Tadmor. New high-resoltion central schemes for nonlinear conseration laws and conection-diffsion eqations. J. Comp. Phs., 60:4 8, 000. [9] H. Li and G. Warnecke. Conergence rates for relaxation schemes approximating conseration laws. preprint. [0] B. Perthame. Boltzmann tpe schemes for gas dnamics and the entrop propert. SIAM J. Nmer. Anal., 7:405 4, 990. [] B. Perthame. Second-order Boltzmann schemes for compressible Eler eqations in one and two space dimensions. SIAM J. Nmer. Anal., 9:, 99. [] Y. Sone. Asmptotic theor of a stead flow of a rarefied gas past bodies for small Kndsen nmbers. In R. Gatignol and Sobbaramaer, editors, Adances in Theor and Continm Mechanics, Proceedings of a Smposim Held in Honor of Henri Cabannes Paris, Springer, pages 9 3,