Hyperbolic Moment Equations Using Quadrature-Based Projection Methods

Transcription

1 Hyperbolic Moment Equations Using Quarature-Base Projection Methos J. Koellermeier an M. Torrilhon Department of Mathematics, RWTH Aachen University, Aachen, Germany Abstract. Kinetic equations like the Boltzmann equation are the basis for various applications involving rarefie gases. An important problem of many approaches since the first evelopments by Gra is the esire global hyperbolicity of the emerging set of partial ifferential equations. Due to lack of hyperbolicity of Gra s moel equations, numerical computations can break own or yiel nonphysical solutions. New hyperbolic PDE systems for the solution of the Boltzmann equation can be erive using quarature-base projection methos.the metho is base on a non-linear transformation of the velocity to obtain a Lagrangian velocity phase space escription in orer to allow for physical aaptivity, followe by a series expansion of the unknown istribution function in ifferent basis functions an the application of quarature-base projection methos.in this paper, we exten the proof for global hyperbolicity of the quarature-base moment system system to arbitrary imensions, utilizing quarature-base projection methos for tensor prouct Hermite basis functions. The analytical computation of the eigenvalues shows the propose corresponence to the Hermite quarature points. Keywors: kinetic equation, hyperbolicity, quarature PACS: Jr, D INTRODUCTION The solution of kinetic equations like the Boltzmann Equations requires an efficient an yet robust iscretization of the velocity space to allow for both physical aaptivity an stability of the numerical metho. The stanar metho to iscretize the resulting equation in velocity space has been evelope by Gra in [] an relies on the expansion of the istribution function in Hermite polynomials. The rawback of this metho is that the resulting PDE system can loose hyperbolicity for certain values of higher moments, see e.g. [2]. In these cases, numerical methos can become unstable because the problem is ill-pose. Existing hyperbolic moels are for example given in [3] where multi-variate Pearson-IV-Distributions are use as an ansatz or in [4] where a maximum entropy approach ensures the hyperbolicity. Another metho by Cai et al. relies on a truncation of the expansion at certain steps of the erivation an has been successfully applie to ifferent problems see [5]. This work has recently been supplemente with a soli theoretical basis in [6], where the proceure is generalize. The novel quarature-base projection methos have been first introuce in [7]. This approach computes the integrals uring the projection by Gaussian quarature which moifies the emerging system an guarantees hyperbolicity for a wie range of cases, provie some simple conitions are fulfille. Up to now there is only a proof of hyperbolicity in the one-imensional case using Hermite polynomials as ansatz functions, but the framework is able to hanle multi-imensional settings as well. The main part of this paper is concerne with the extension of the proof given in [8] to arbitrary imensions an more flexible basis expansions. The evelope framework can be applie to this setting analogously to the one-imensional case as we will show. This results in a straightforwar proof an makes it possible to calculate the eigenvalues of the system analytically. The paper is organize as follows: We first explain the basic concepts an recall the equations erive in [8] using quarature-base projection methos, along with the conitions for hyperbolicity of the emerging system. A proof for global hyperbolicity of the PDE system in the general -imensional case is etaile in the main section. The paper ens with a short conclusion. Proceeings of the 29th International Symposium on Rarefie Gas Dynamics AIP Conf. Proc. 628, ; oi: 0.063/ AIP Publishing LLC /$

2 TRANSFORMED BOLTZMANN TRANSPORT EQUATION We consier the Boltzmann equation without external force for the evolution of the mass ensity function f t,x,c t f t,x,c+c i f t,x,c=s f, x i where we assume a -imensional setting, i.e. x R an velocity v R. Inex notation is use whenever the inices are no further specifie, e.g. in the transport term of Equation. The collision operator S f on the right-han sie will be neglecte throughout this paper as we focus on the transport part of Equation. Moels for the right-han sie can be foun in [9], [0] an []. The macroscopic quantities ensity ρt, x, velocity vt, x an energy θt, x of the istribution function can be compute by integration over the velocity space we will omit the arguments most of the time for better reaability. ρ = f t,x,c c, ρv = c f t,x,c c, ρθ = c v 2 f t,x,c c. 2 R R R Multiplication of Eq. with,c,c 2 i followe by integration over the velocity space yiels the conservation laws of mass, momentum an energy, see e.g. [2]. We recall the transforme version of the Boltzmann Eq. that essentially uses a Lagrangian velocity phase space an thus exhibits physical aaptivity which allows for efficient an yet simple iscretizations, introuce also in [3]. As escribe in [7], the velocity is transforme in a highly non-linear way to allow for intrinsic physical aaptivity of the scheme. We shift the microscopic velocity c by its macroscopic counterpart v an scale by the stanar eviation θ. The corresponing Galilei-invariant variable transformation reas c c vt,x θt,x =: ξ t,x,c. 3 Every Maxwellian in c-space is thus transforme to a Gaussian in ξ -space scale with ensity ρ, as epicte in [7]. We furthermore use a scale istribution function f t,x,ξ, efine by f t,x,ξ = ρ θ f t,x,ξ. 4 Accoring to [7] an with the help of the convective time erivative D t := t + v i xi, Equation transforms to ρ D tρ 2θ D tθ f + D t f + θξ j ρ x j ρ 2θ x j θ f + x j f + ξ j f D t v j + θξ i xi v j θ 2θ ξ j D t θ + θξ i xi θ 5 = 0. T This PDE for the vector of unknowns u = ρ,v,...,v,θ, f can be written in the following compact notation Λ AD t u + Λ θξ i A xi u = 0, 6 using the efinitions A = f, f,..., f f, f ξ j, R +3, 7 ξ ξ ξ j Λ = ρ θ I 2 θ R

3 Equation 6 is uneretermine, as ρ,v an θ are not known. But the efinition of the macroscopic variables accoring to Eq. 2 leas to + 2 aitional algebraic equations that formally close the system. It was shown in [8] that point evaluations of Eq. 6 together with a reasonable finite ifference approximation of the erivatives ξ j f o in general not lea to a globally hyperbolic system as eigenvalues of the system matrix epen on the unknowns. A eficiency that is remove with the help of quarature-base projection methos, as explaine in the following Section. GENERAL FRAMEWORK FOR HYPERBOLIC APPROXIMATION OF KINETIC EQUATIONS We expan the unknown istribution function f t, x, ξ aroun its equilibrium Maxwellian with respect to basis functions φ α ξ an corresponing coefficients κ α t,x, obtaining f t,x,ξ = n α= κ α t,xφ α ξ. 9 with ifferentiable ansatz functions φ α ξ. After insertion of Ansatz 9 into Eq. 6, we multiply with test functions ψ β ξ for β =,...,n an integrate over the velocity space to obtain a system of equations for the unknowns ρ,v,θ an the basis coefficients κ α. This can be interprete as a projection with projection operator P P β g := g,ψ β = gψ R β ξ. 0 As the efinition of P using exact integration oes not lea to global hyperbolic systems see [5], we use an approximative projection operator Q, that computes the integrals by Gauss-quarature P β g Q β g := n k= w k gξ k ψ β ξ k. with positive quarature weights w k an pairwise istinct quarature points ξ k R, for k =,...,n. Applying Q β for β =,...,n to Eq. 6 with Expansion 9 inserte, the emerging PDE system can be written in the following form Ψ T WAΛ D Dt ũ + Ψ T WΞ i AΛ θ ũ = 0, 2 i= x i with unknowns ũ =ρ,v,...,v,θ,κ 0,...,κ n T R n++2, 3 a columnwise efine system matrix A = Φκ, κ,..., κ, Φκ Ξ j κ,φ R n n++2, 4 ξ ξ ξ j consisting of matrices for i, j =,...,n an k =,..., Ψ i, j = ψ j ξ i, Φ i, j = φ j ξ i, Ξ k i, j = ξ i ξ k i, j = ξk φ j ξi 5 δ ij, k W i, j = ω i δ ij 6 as well as Λ from Eq. 26 with I n instea of as last entry. It is alreay clear, that System 2 is not close, as its n equations inclue n unknowns. A moel reuction is explaine in the following accoring to the proceure escribe in [8]. 628

4 Inserting the scale istribution function from Eq. 4 an using 9, we get + 2 compatibility conitions from Eqs. 2 = κ α φ α ξ,, 0 = κ α φ α ξ,ξ i, for i =,...,, = κ α φ α ξ,ξi 2. 7 The compatibility conitions 7 can be use to reuce the number of unknowns in the PDE system to obtain a close PDE system similar to the general proceure escribe in [4]. We thus use Q R l n,c =,0,...,0, T R l an write the l = + 2 Eqs. 7 in matrix-vector form to efine Q by Qκ = c. 8 Now we ecompose Q R l n as follows e.g. using singular value ecomposition: Q = SMT, with regular S R l l, M = Q,0 R l n, regular Q R l l an regular T R n n. 9 We can then solve for the first l variables of a transforme set of variables β R n using β = β0 β an then close the system of PDEs by inserting β0 β 0 κ = T = T β,t 2 β := T κ β 0 = Q S c 20 = T Q S c + T 2 β, 2 with a proper splitting of the columns of T = T,T 2. With the help of the compatibility conitions, the set of n basis coefficients κ is thus reuce to a set of n l variables β which leas to a close PDE system with the same number of equations an unknowns. Rewriting System 2 with the help of the reuce variables, we get Ψ T D WA β Λ β Dt u + Ψ T WΞ i A β Λ β θ u = 0 22 i= x i with a columnwise efine system matrix A β of which the first + 2 columns inclue the transforme coefficients β: A β = Φ β,,...,, Φ ξ β ξ β Ξ j,φt β ξ 2 R n n. 23 j β Aitionally, we use the efinition of the matrices for i, j =,...,n an k =,..., Φ β = Φ T Q S c + T 2 β, 24 = T Q S c + T 2 β, 25 ξ k β ξ k Λ β = ρ θ I 2 θ I n 2 Rn n. 26 The subscript β thus enotes a epenence of the corresponing matrix on the unknowns β. The important question of hyperbolicity of System 22 is aresse by Theorem in [8] an requires essentially only the regularity of matrices Ψ,W,Λ β an A β, of which the regularity of matrix A β is the only non-trivial task for most cases. 629

5 Provie these requirements, the generalize eigenvalues λ k for k =,...,n are all real an rea λ k = i= n i ξ k i θ + vi, for k =,...,n. 27 After the valiation of the conitions for the one-imensional case in previous papers, we will show global hyperbolicity in the general -imensional case in the following Section. GLOBALLY HYPERBOLIC SYSTEM USING HERMITE ANSATZ AND QUADRATURE IN D DIMENSIONS A proof of global hyperbolicity for the one-imensional system with Hermite ansatz functions can be foun in [8]. We will now exten the proof to the general case of imensions in position an velocity space, i.e. x,ξ R. We consier the following basis expansion for the scale istribution function f in orthonormal basis functions f t,x,ξ = κ α t,xφ α ξ, 28 α i N i for a -imensional multi-inex α N an a vector N =N 0,...,N N specifying the moment theory. The total number of unknown basis coefficients is enote as N = i= N i +. The corresponing basis functions are the tensor prouct of one-imensional Hermite functions φ α ξ = i= φ αi ξ i, φ i x=he i x e x2 /2 2π an test functions ψ α ξ = i= He i ξ i, 29 with normalize Hermite polynomial He i of egree i such that the following recursion formulas hol α = α α i + Φ α+ei, ξ i = α i + αi + 2Φ α+2ei + α i + Φ α. 30 ξ i ξ i Note that this ansatz 28 an 29 exhibits some flexibility, because ifferent choices for N i can be chosen to capture the ifferent behaviour of the istribution function in each spatial irection. We will nevertheless loose rotational invariance with this separation ansatz. The quarature projection is one using multi-imensional Gauss-Hermite quarature as follows R gξ ξ N i =0... N i =0 g ξ i,...,ξ i w,i...w,i, 3 using quarature points ξ i j an weights w i, j given by ξ i j = j-th root of He Ni + ξ i, for i =,...,, j = 0,...,N i 32 w i, j = j-th weight of quarature formula in irection i, for i =,...,, j = 0,...,N i 33 A compact notation for the quarature formula is N gξ ξ R g ξ k w k, k= 34 with the total number of quarature points equal to the number of basis coefficients N an a consecutive orering of the quarature points accoring to ξ k { ξ i,...,ξ i i j {0,...,N j }, j =,...,} 35 w k {Π j=w j,i j i j {0,...,N j }, j =,...,} 36 The setting escribe above enables us to prove the following central theorem of this paper: Theorem Using tensor prouct Hermite basis an test functions as specifie in Eqs. 29 together with Gauss- Hermite quarature, the conitions of Theorem in [8] are fulfille an the resulting PDE system is globally hyperbolic. 630

6 Proof We first reuce the number of unknowns as escribe in using the ecomposition 9 an state the compatibility conitions erive using Equation 7 as = κ 0, 0 = κ ei, for i =,...,, = κ i= κ 2ei. 37 We orer the basis coefficients introucing the inex set N κ = {α N < α,α i N i,α 2e i,i =,...,} κ = κ 0,κ ei,κ 2ei, κ α α N κ. 38 to write Equations 37 in matrix vector form as Qκ = c, see 8, with c =,0,...,0, T, Q = A ecomposition Q = SMT is one accoring to Equation 9 using I }{{} 39 I + S =......, M =... 0, T =... 2 }{{} I N 2 2 }{{} Q 40 Splitting the matrix T into two parts as propose in Eq. 2, we can ientify the multiplication of a matrix with T 2 from the right as eletion of the first + columns an subtraction of the + 2 n column from columns + 3upto Multiplication with T extracts the first + 2 columns. The unknown vector κ can be rewritten with only N 2 variables as κ =,0,...,0, i= κ 2ei, κ α α N κ which correspons to the solution of Eqs. 37 for the first + 2 unknowns. T, 4 We will now erive matrix A β in 23 columnwise in orer to show regularity of A β later. For better reaability, we introuce a set of multi-inices N β0 = {0,e,...,e,2e }, such that the corresponing basis coefficients κ α for α N β0 are all the coefficients β 0, that we have explicitly calculate using the compatibility conitions, i.e. the first + 2 entries in κ an the remaining multi-inices form the set N β = {α N,α i N i, < α,α 2e }. The last N 2 columns of A β see 23 for the efinition, ientifie by the corresponing multi-inex α N β, are ΦT 2 an rea ΦT 2α = Φ α Φ 2e δ α,2e j, for α N β. 42 j=2 The first column Φ β can be written as Φ β = Φ 0 + j=2 Φ 2e j Φ 2e κ 2e j + α N κ Φ α κ α 43 63

7 Columns 2 to + can be simplifie using the recursion formulas in 30 = Φ ξ ei 2Φ 2e +e i i κ 2e j + Φ αi α+ei + κ α, 44 β j=2 α N β,α i <N i where the last sum only inclues α i < N i, because a value α i = N i woul evaluate Φ α+ei to zero, as the i th entry of each quarature point is a root of the N i + st Hermite polynomial. For the remaining + 2 n column, the term j= ξ j can be written as ξ j β j= ξ j ξ j β so that the entire column reas ξ j ξ j β = j= j= α j <N j α j + Φ α κ α α j =N j Φ α κ α + j= α j +<N j α j + α j + 2Φ α+2e j κ α, 45 Φ β = α j Φ α κ α + α j + α j + 2Φ α+2e j κ α j= α j <N j α j +<N j 46 = 2 Φ 2e j + 2 Φ 2e j Φ 2e κ 2e j + Φ α γ α, j= j=2 α N κ 47 for γ α R incluing some coefficients κ α to sum up the higher orer terms. With the help of the expressions for the single columns, we can now easily go through the conitions of Theorem in [8] an prove the central statement of this paper. We have to check regularity of matrices Ψ,W,Λ β an A β in orer to verify global hyperbolicity of System 22 in our setting. Ψ is regular by construction, as its columns are point evaluations of each orthogonal test function. W is a regular iagonal matrix, because the quarature weights are strict positive for multi-imensional Gauss- Hermite quarature. Λ β is regular, if an only if ρ 0 an θ 0, this is true by assumption of Ansatz 4. As erive above, every column of A β is a linear combination of some Φ α. Linear inepenence of its colums can be prove by checking where each column Φ α appears. In total, we must not have more than N columns Φ α involve an all columns of A β have to be linearly inepenent of each other. ΦT 2 see Eq. 42 has linear inepenent columns Φ α Φ 2e j=2 δ α,2e j, for α N β, 2 Φ β see Eq. 43 is a linear combination of columns of ΦT 2 plus an aitional Φ 0 an thus linearly inepenent of ΦT 2, 3 see Eq. 44 is a linear combination of the columns of ΦT ξ i 2 plus an aitional Φ β ei an thus linearly inepenent of the others 4 Φ β j= ξ j see Eq. 47 is a linear combination of columns of ΦT ξ j 2 plus an aitional 2 j= Φ 2e β j an thus linearly inepenent of the others. Accoring to Eq.27, the eigenvalues of the system matrix can be erive analytically an are closely connecte with the Hermite roots. The following two-imensional example in Fig. shows that the eigenvalues in fact lie on circles going through the origin an one of the quarature points, which consists of Hermite roots. CONCLUSION After the presentation of the most important notations an previous results, we have escribe a very flexible ansatz using tensor prouct Hermite basis functions of ifferent orer in each spatial irection. This ansatz is capable of capturing the important features of the istribution function, e.g. for applications with certain pronounce irections. 632

8 y 2 2 y 2 2 x 2 2 x 2 2 FIGURE. Eigenvalue plot of 2D example for asymmetric N = 3,N 2 = left an symmetric N = N 2 = 2 right By extension of the proof for hyperbolicity from the one-imensional case to the multi-imensional case, we have shown another successful application of the general framework evelope in [8]. With the help of simple quarature rules an their properties, the necessary expressions coul be erive an it was possible to show global hyperbolicity of the system of equations. Furthermore, the eigenvalues of the generalize system matrix coul be compute analytically. ACKNOWLEDGMENTS This work has been carrie out with the support of the German National Acaemic Founation. REFERENCES. H. Gra, Comm. Pure Appl. Math. 2, F. Brini, Continuum Mech. Thermoyn. 3, M. Torrilhon, Comm. Comput. Phys. 7, C. D. Levermore, Journal of Statistical Physics 83, Z. Cai, Y. Fan, an R. Li, Commun. Math. Sci., Z. Cai, Y. Fan, an R. Li, "A Framework on Moment Moel Reuction for Kinetic Equation," submitte. 7. J. Koellermeier, Hyperbolic Approximation of Kinetic Equations Using Quarature-Base Projection Methos, Master s thesis, RWTH Aachen University J. Koellermeier, R. Schaerer, an M. Torrilhon, Kinet. Relat. Mo, 7, P. L. Bhatnagar, E. P. Gross, an M. Krook, Physical Review 94, C. Cercignani, The Boltzmann Equation an its Application, Springer, S. Heinz, Statistical Mechanics of Turbulent Flows, Springer, H. Struchtrup, Macroscopic Transport Equations for Rarefie Gas Flows, Springer, P. Kauf, Multi-Scale Approximation Moels for the Boltzmann Equation, Ph.D. thesis, ETH Zürich E. Hairer, an G. Wanner, Solving Orinary Differential Equations II, Stiff an Differential-Algebraic Problems, Springer Verlag,