Hyperbolic Moment Equations Using Quadrature-Based Projection Methods

Transcription

1 Using Quadrature-Based Projection Methods Julian Köllermeier, Manuel Torrilhon 29th International Symposium on Rarefied Gas Dynamics, Xi an July 14th, 2014 Julian Köllermeier, Manuel Torrilhon 1 / 12

2 Outline 1 Transformed Boltzmann Transport Equation Julian Köllermeier, Manuel Torrilhon 2 / 12

3 Boltzmann Transport Equation Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions Julian Köllermeier, Manuel Torrilhon 3 / 12

4 Boltzmann Transport Equation Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions t f (t, x, c) + c i f (t, x, c) = S(f ) x i PDE for particles probability density function f (t, x, c) Describes change of f due to transport and collisions Collision operator S x R d, c R d No external force Julian Köllermeier, Manuel Torrilhon 3 / 12

5 Transformation of Velocity Variable Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions f c ξ(t, x, c) := c v(t, x) θ(t, x) = f (ξ) = ρ ) exp ( ξ2 2πθ 2 10 f (c) = ρ ) (c v)2 exp ( 2πθ 2θ c Julian Köllermeier, Manuel Torrilhon 4 / 12

6 Transformation of Velocity Variable Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions f c ξ(t, x, c) := c v(t, x) θ(t, x) = f (ξ) = ρ ) exp ( ξ2 2πθ 2 10 f (c) = ρ ) (c v)2 exp ( 2πθ 2θ c Lagrangian velocity space reduces numerical complexity Julian Köllermeier, Manuel Torrilhon 4 / 12

7 Transformation of Velocity Variable Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions f c ξ(t, x, c) := c v(t, x) θ(t, x) = f (c) = ρ ) (c v)2 exp ( 2πθ 2θ c f (ξ) = ρ ) exp ( ξ2 2πθ 2 Lagrangian velocity space reduces numerical complexity Julian Köllermeier, Manuel Torrilhon 4 / 12

8 Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions Transformed Boltzmann Transport Equation Transformation: ξ(t, x, c) := c v(t,x) θ(t,x) Scaled distribution function: f (t, x, ξ) = ρ θ d f (t, x, ξ) Julian Köllermeier, Manuel Torrilhon 5 / 12

9 Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions Transformed Boltzmann Transport Equation Transformation: ξ(t, x, c) := c v(t,x) θ(t,x) Scaled distribution function: f (t, x, ξ) = ρ θ d f (t, x, ξ) ( 1 ρ D tρ d ) 2θ D tθ f + Dt f + (( 1 θξj ρ x j ρ d ) ) 2θ x j θ f + xj f + ξj f ( θ 1 (D t v j + ) θξ i xi v j 1 ( 2θ ξ j D t θ + θξ i xi θ) ) = 0. Julian Köllermeier, Manuel Torrilhon 5 / 12

10 Compatibility Conditions Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions Macroscopic variables ρ, v, θ are moments of the PDF f ρ(t, x) = ρ(t, x)v(t, x) = d ρ(t, x)θ(t, x) = f (t, x, c) dc, R d cf (t, x, c) dc, R d c v 2 f (t, x, c) dc. R d Julian Köllermeier, Manuel Torrilhon 6 / 12

11 Compatibility Conditions Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions Macroscopic variables ρ, v, θ are moments of the PDF f 1 = 0 = d = f (ξ) dξ, R d ξ f (ξ) dξ, R d ξ 2 f i (ξ) dξ, R d Julian Köllermeier, Manuel Torrilhon 6 / 12

12 Compatibility Conditions Boltzmann Transport Equation Transformation of Velocity Variable Compatibility Conditions Macroscopic variables ρ, v, θ are moments of the PDF f Ansatz 1 = f (ξ) dξ, R d 0 = ξ f (ξ) dξ, R d d = ξ 2 f i (ξ) dξ, R d n f (t, x, ξ) = α i (t, x)φ i (ξ) i=0 Julian Köllermeier, Manuel Torrilhon 6 / 12

13 Hyperbolicity Transformed Boltzmann Transport Equation Hyperbolicity Failure of Classical Methods Hyperbolic Approximation Julian Köllermeier, Manuel Torrilhon 7 / 12

14 Hyperbolicity Transformed Boltzmann Transport Equation Hyperbolicity Failure of Classical Methods Hyperbolic Approximation Definition: Hyperbolicity The PDE system of the form t u + A(u) x u = 0 is globally hyperbolic if matrix A is diagonalizable with real eigenvalues for every u. Hyperbolicity necessary for Well-posedness of the initial value problem and stable solutions Physical solutions with real-valued propagation speed Julian Köllermeier, Manuel Torrilhon 7 / 12

15 Failure of Classical Methods Hyperbolicity Failure of Classical Methods Hyperbolic Approximation Discrete Velocity Method Ansatz: Discrete point evaluations of transformed Boltzmann Equation Projection: δ(ξ ξ j )dξ R d Julian Köllermeier, Manuel Torrilhon 8 / 12

16 Failure of Classical Methods Hyperbolicity Failure of Classical Methods Hyperbolic Approximation Discrete Velocity Method Ansatz: Discrete point evaluations of transformed Boltzmann Equation Projection: δ(ξ ξ j )dξ R d Grad s Method [Grad, 1949] Ansatz: f (t, x, ξ) = 1 ) n exp ( ξ2 α i (t, x)he i (ξ) 2π 2 i=0 Projection: He j (ξ)dξ R d Julian Köllermeier, Manuel Torrilhon 8 / 12

17 Failure of Classical Methods Hyperbolicity Failure of Classical Methods Hyperbolic Approximation Discrete Velocity Method Ansatz: Discrete point evaluations of transformed Boltzmann Equation Projection: δ(ξ ξ j )dξ R d Grad s Method [Grad, 1949] Ansatz: f (t, x, ξ) = 1 ) n exp ( ξ2 α i (t, x)he i (ξ) 2π 2 i=0 Projection: He j (ξ)dξ R d Methods are not globally hyperbolic, imaginary eigenvalues Julian Köllermeier, Manuel Torrilhon 8 / 12

18 Hyperbolic Approximation Hyperbolicity Failure of Classical Methods Hyperbolic Approximation Quadrature-based Projection Method [Koellermeier et al., 2014] Ansatz: f n (t, x, ξ) = α i (t, x)φ i (ξ) Projection: R d i=0 Ψ j (ξ)dξ n w k ξk Ψ j (ξ k ) k=0 Julian Köllermeier, Manuel Torrilhon 9 / 12

19 Hyperbolic Approximation Hyperbolicity Failure of Classical Methods Hyperbolic Approximation Quadrature-based Projection Method [Koellermeier et al., 2014] Ansatz: f n (t, x, ξ) = α i (t, x)φ i (ξ) Projection: R d Global Hyperbolicity i=0 Ψ j (ξ)dξ n w k ξk Ψ j (ξ k ) k=0 PDE system is globally hyperbolic, eigenvalues can be precomputed under some simple conditions on Φ i, Ψ i, ξ k and w k Julian Köllermeier, Manuel Torrilhon 9 / 12

20 Hermite Basis and Gauss-Hermite Quadrature Comparison to Grad and Cai in 1D Hermite Basis and Gauss-Hermite Quadrature Julian Köllermeier, Manuel Torrilhon 10 / 12

21 Hermite Basis and Gauss-Hermite Quadrature Comparison to Grad and Cai in 1D Hermite Basis and Gauss-Hermite Quadrature Quadrature-based Projection Method Grad ansatz: Φ i (ξ) = 1 2π e ξ/2 He i (ξ), Ψ i (ξ) = He i (ξ) Gauss-Hermite quadrature: Points ξ k are roots of He N+1 (ξ), w k are corresponding weights Julian Köllermeier, Manuel Torrilhon 10 / 12

22 Hermite Basis and Gauss-Hermite Quadrature Comparison to Grad and Cai in 1D Hermite Basis and Gauss-Hermite Quadrature Quadrature-based Projection Method Grad ansatz: Φ i (ξ) = 1 2π e ξ/2 He i (ξ), Ψ i (ξ) = He i (ξ) Gauss-Hermite quadrature: Points ξ k are roots of He N+1 (ξ), w k are corresponding weights Global Hyperbolicity Global hyperbolicity has been analytically proven in 1D and general d-dimensional case Eigenvalues contain roots of Hermite polynomials: λ k = θ ξ k + v Julian Köllermeier, Manuel Torrilhon 10 / 12

23 Comparison to Grad and Cai in 1D Hermite Basis and Gauss-Hermite Quadrature Comparison to Grad and Cai in 1D Grad s Method [Grad, 1949] Ansatz: Expansion in Hermite ansatz functions Projection: Exact integration Julian Köllermeier, Manuel Torrilhon 11 / 12

24 Comparison to Grad and Cai in 1D Hermite Basis and Gauss-Hermite Quadrature Comparison to Grad and Cai in 1D Grad s Method [Grad, 1949] Ansatz: Expansion in Hermite ansatz functions Projection: Exact integration Cai s Method [Cai et al. 2014] Ansatz: Arbitrary basis function Regularization: Cut-off higher terms during computations Additional terms only in last equation Julian Köllermeier, Manuel Torrilhon 11 / 12

25 Comparison to Grad and Cai in 1D Hermite Basis and Gauss-Hermite Quadrature Comparison to Grad and Cai in 1D Grad s Method [Grad, 1949] Ansatz: Expansion in Hermite ansatz functions Projection: Exact integration Cai s Method [Cai et al. 2014] Ansatz: Arbitrary basis function Regularization: Cut-off higher terms during computations Additional terms only in last equation Quadrature-based projection Method [Koellermeier et al., 2014] Ansatz: Arbitrary basis function Regularization: Corresponding Gaussian-quadrature rule Additional terms in last two equations, includes Cai s regularisation terms Julian Köllermeier, Manuel Torrilhon 11 / 12

26 Summary and Further Work Summary and Further Work Julian Köllermeier, Manuel Torrilhon 12 / 12

27 Summary and Further Work Summary and Further Work Results Development of abstract framework Derivation of hyperbolicity conditions and eigenvalues General proof for hyperbolicity of d-dimensional system Julian Köllermeier, Manuel Torrilhon 12 / 12

28 Summary and Further Work Summary and Further Work Results Development of abstract framework Derivation of hyperbolicity conditions and eigenvalues General proof for hyperbolicity of d-dimensional system Summary Quadrature-based projection methods derive globally hyperbolic PDE systems from the transformed Boltzmann Equation Julian Köllermeier, Manuel Torrilhon 12 / 12

29 Summary and Further Work Summary and Further Work Results Development of abstract framework Derivation of hyperbolicity conditions and eigenvalues General proof for hyperbolicity of d-dimensional system Summary Quadrature-based projection methods derive globally hyperbolic PDE systems from the transformed Boltzmann Equation Further Work Different expansions, basis functions, quadrature formulas Use of sparse-tensor product approximations Numerics for the (non-conservative) hyperbolic PDE system Julian Köllermeier, Manuel Torrilhon 12 / 12

30 References Transformed Boltzmann Transport Equation Summary and Further Work J. Koellermeier, R.P. Schaerer and M. Torrilhon A Framework for Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods, to appear in Kin. Rel. Mod. 7(3) (2014) J. Koellermeier, M. Torrilhon Using Quadrature-Based Projection Methods, 29th Rarefied Gas Dynamics, Xi an Z. Cai, Y. Fan and R. Li Globally hyperbolic regularization of Grad s moment system, Comm. Pure Appl. Math., 67(3) (2014), H. Grad On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2(4) (1949), Julian Köllermeier, Manuel Torrilhon 13 / 12

31 Hermite Polynomials 1D Summary and Further Work He n (ξ) := 1 n! Hen (ξ), for n N. Orthonormality w.r.t. the scalar product: φ, ψ w = Recursion formulas: + φ(ξ)ψ(ξ)w(ξ)dξ, for w(ξ) := 1 2π e ξ2 /2. ξhe n (ξ) = n + 1He n+1 (ξ) + nhe n 1 (ξ), He n(ξ) = nhe n 1 (ξ), d ( ) He n (ξ)w(ξ) = w(ξ) n + 1He n+1 (ξ), dξ ξ d ( ) He n (ξ)w(ξ) = w(ξ) n n + 1( + 2Hen+2 (ξ) + ) n + 1He n (ξ). dξ Julian Köllermeier, Manuel Torrilhon 14 / 12

32 Multi-dimensional Expansion Summary and Further Work f (t, x, ξ) = α i N i κ α (t, x)φ α (ξ), Basis functions are tensor product of 1D Hermite functions: d φ α (ξ) = φ αi (ξ i ), i=1 Recursion formulas: 1 φ i (x) = He i (x) e x2 /2 2π d ψ α (ξ) = He i (ξ i ), i=1 Φ α ξ i = α i + 1Φ α+ei, ξ i Φ α ξ i = α i + 1 ( α i + 2Φ α+2ei + α i + 1Φ α ). Julian Köllermeier, Manuel Torrilhon 15 / 12

33 Gauss-Hermite Quadrature Summary and Further Work Multi-dimensional Gauss-Hermite quadrature N 1 g(ξ)dξ R d i 1 =0 N d i d =0 Quadrature points (ξ i ) j and weights w i,j : (ξ i ) j = j-th root of He Ni +1 (ξ), g ((ξ 1 ) i1,..., (ξ d ) id ) w 1,i1... w d,id, w i,j = j-th weight of quadrature formula in direction i. Compact notation for quadrature formula R d g(ξ)dξ N g (ξ k ) w k. k=1 Julian Köllermeier, Manuel Torrilhon 16 / 12

34 Hyperbolicity Conditions Summary and Further Work Emerging PDE system after compatibility conditions: Ψ T D d WA β Λ β Dt u + Ψ T WΞ i A β Λ β θ u = 0 (1) x i Hyperbolicity conditions: i=1 (1) Regularity of matrix Ψ, (2) Regularity of matrix W or equivalently ω i 0, i = 0,..., n, (3) Regularity of matrix Λ β, (4) Regularity of matrix A β or equivalently ranka β = n + 1. (2) Julian Köllermeier, Manuel Torrilhon 17 / 12