A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

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1 Work supported by NSF grants DMS and DMS Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21 A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit Jeff Haack Department of Mathematics, and Institute for Computational Engineering Science, University of Texas at Austin Joint work with with Irene M. Gamba (UT) Issues in Solving the Boltzmann Equation for Aerospace Applications ICERM, Providence RI 7 June 213

2 Outline of talk 1 Spectral method for Boltzmann-type collision operators with anisotropic scattering Parallelization 2 Grazing collisions and the Landau equation Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

3 The Boltzmann equation Boltzmann equation: Df Dt (x, v, t) = Q(f, f )(v), v R3, Q(f, f ) is the collision operator: Q(f, f )(v) = B( u, û σ)(f (v )f (v ) f (v )f (v))dσdv R 3 S 2 Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

4 Numerical methods Direct Simulation Monte Carlo Bird, Nanbu,... conservation, positivity, overcomes dimensionality noise, transients, time dependent problems, tails, low speed flows, etc. Deterministic methods (spectral, DVM...) Advantages: no noise, accuracy in v Disadvantages: Positivity, conservation, dimensionality Previous spectral works Bobylev (75), Bobylev-Rjasanow (97, 99, ), Ibragimov-Rjasanow (2), Pareschi-Russo (), Mouhot-Pareschi (4), Pareschi-Russo-Toscani () (Landau), Pareschi-Toscani-Villani (3) (Grazing collisions consistency) Weak form: Gamba-Tharkabhushanam (9, 1) - conservation, inelastic Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

5 Spectral formulation of collision operator Weak (Maxwell) form of collision operator: Q(f, f ) φ(v) dv = f (v)f (v )[φ(v ) φ(v)] B( u, cos θ) dσdv dv, Let φ(v) = e iζ v /( 2π) 3, then we have that the Fourier transform of the collision integral is Q(ζ) = = = F{f (v)f (v u)}(ζ)g(ζ, u)du R 3 1 G(u, ζ) u R 3 ( ˆf (ζ ξ)ˆf (ξ)e iξ u dξdu 2π) 3 ξ R 3 1 ( ˆf (ζ ξ)ˆf (ξ)ĝ(ξ, ζ)dξ 2π) 3 ξ R 3 Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

6 Spectral formulation of collision operator The convolution weights G(ζ, u), Ĝ(ξ, ζ) are given by G(ζ, u) = B( u, cos θ)(e i ζ 2 (u u) 1)dσ. σ S 2 Ĝ(ξ, ζ) = e iξ u B( u, cos θ)(e i ζ 2 (u u) 1)dσdu. u R 3 σ S 2 For B = u λ /4π (isotropic), this can be reduced to Ĝ(ξ, ζ) = no time dependence r λ+2 [sinc( r ζ 2 )sinc(r ξ ζ ) sinc(r ξ )]dr 2 Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

7 Difficulties in simulating the Boltzmann equation Issues: Q(f, f )(v) = R 3 S 2 B( u, û σ)(f (v )f (v ) f (v )f (v))dσdv Dimensionality: requires O(N 6 ) operations for a single evaluation of weighted convolution Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

8 Computation is embarassingly parallel! Test problem: 128 spatial points, 24 3 velocity mesh points Time listed: wall time using Stampede (NSF XSEDE resource) for one timestep ( 244 billion operations) nodes cores time (s) [H. submitted 213, also arxiv] Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

9 Sudden Heating problem.5.4 t/t =.5 x 1 /x =.1 x 1 /x =.3 x 1 /x =.5 x 1 /x =.9 g(v 1 ) v 1 /v Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

10 Difficulties in simulating the Boltzmann equation Q(f, f )(v) = B( u, û σ)(f (v )f (v ) f (v )f (v))dσdv R 3 S 2 Issues: Dimensionality: requires O(N 6 ) operations for a single evaluation of weighted convolution Conservation: collision invariants need to be preserved. Q(f, f ) v R 3 1 v v 2 =. Solve the constrained minimization problem in O(N 3 ) (Gamba, Tharkabhushanam) { min 1 } 2 Q Q 2 2 CQ = Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

11 Anisotropic scattering Potentials interactions (e.g. Coulomb) include an angular component. Write B = u λ b(cos θ). In this case, the untransformed weight G(u, ζ) is given by G(u, ζ) = u λ ( b(û σ)(e i ζ S d 1 π = 2π u λ b(cos θ) sin θ ζ = ζ (ζ u/ u )u/ u. L π Ĝ(ζ, ξ) = 4π 2 r λ+2 π 2 u e ( (1 cos θ)ζ i e i ζ u 2 σ 1)dσ ) ( u sin θ ζ 2 u J 2 ) ) 1 dθ, ) b(cos θ) sin θ sin φj (r ξ sin φ [ ( cos r(ξ ζ ) ( ) 2 (1 cos θ)) ζ 12 ζ cos φ J r ζ sin φ sin θ ( ) ] ζ cos rξ ζ cos φ dθdφdr Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

12 The Coulombic (grazing) limit It is well known that when the underlying potential is Coulombic, long distance grazing collisions dominate the collision term (Landau 36). ( Q L = v u λ+2 (δ ij u ) iu j u 2 )(f (v ) v f (v) f (v) v f (v )) dv Similar weak formulation gives G L (u, ζ) = u λ ( 4i(ζ u) u 2 ζ 2) Pareschi, Toscani, and Villani (23): convergence to Landau for collocation. However, no computations were done in this limiting regime to our knowledge. Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

13 Approximating Boltzmann by Landau Using Screened Coulomb: B = u 3 C sin 4 (θ/2) 1 θ ε, ε r /λ 3 D. Isotropic initial condition: ( f (v, ) = 1 1 e 1 ε = 1 4, N = 16 ( ) ) v f(v 1 ) x v 1 Solid lines: Landau solution. Dashed lines: Boltzmann solution. t =, 1, 2, 5, 1. Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

14 Scattering kernel Make the following assumptions on the scattering kernel. Let ε > be the small parameter associated with the grazing collision limit. A family of kernels b ε are grazing if (Villani, Bobylev): Λ ε = 2π π b ε(cos θ) sin 2 (θ/2) sin θdθ Λ < θ >, b ε (cos θ) uniformly on θ θ This corresponds to π b ε(cos θ) sin θ θ k dθ, k > 2 Some examples Note for ε-linear: π sin θ Coulomb: b ε (cos θ) sin θ = C sin 4 (θ/2) log sin(ε/2) 1 θ ε ε-linear: b ε (cos θ) sin θ = 8ε πθ 4 1 θ ε, b ε (cos θ) sin θ(θ 2 )dθ Cε θ π = C(1 ε ε π ) Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

15 Convergence of Boltzmann to Landau Q ε (ζ) = R 3 F{f ε (v)f ε (v u)}(ζ)g ε (ζ, u)du Theorem (H., Gamba.) Assume that f ε satisfies F{f ε (v)f ε (v u)}(ζ) A(ζ) 1 + u 3, (1) with A uniformly bounded in ζ, and that b(cos θ) is the screened Rutherford cross section. Then the rate of convergence of the Boltzmann collision operator with grazing collisions to the Landau collision operator is given by Q L [f ε ](ζ) Q bε [f ε ](ζ) O ( ) 1 log sin(ε/2) as ε. (2) Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

16 sketch of proof ( G ε (u, ζ) = u λ b ε (cos θ)(e i ζ S d 1 = u λ π ( 2π b ε (cos θ) sin θ 2 u e ) ζ u i 2 σ 1)dσ e i 2 ((1 cos θ)ζ u+ u ζ j sin θ sin φ+ u ζ k sin θ cos φ) 1 ) dφdθ, First two terms of the expansion: = 2 u 3 log sin(ε/2) π = u 3 ( 4i(u ζ) u 2 ζ 2) ε cos(θ/2) sin(θ/2) i(u ζ) 1 2 sin(θ/2) cos(θ/2)(u ζ)2 dθ u 3 ( 1 2 (u ζ) ζ 2 u 2 ) (1 + cos ε) log sin(ε/2). Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

17 sketch of proof Q bε [f ε ](ζ) = Q L [f ε ](ζ) + F{f ε (v)f ε (v u)}(ζ)r(ζ, u)du (3) R n = Q L [f ε ](ζ) + F{f ε (v)f ε (v u)}(ζ) R n + ( u 3 log sin(ε/2) u 3 log sin(ε/2) ( (u ζ) ) 2 4 ζ 2 u 2 (1 + cos ε) ) G bε,n(ζ, u) du. n=3 Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

18 sketch of proof R(ζ, u) 2 ζ 2 (1 + cos ε) u log sin(ε/2) + 2π ζ 3 log sin(ε/2) All integrable at zero, and gives integrability at ( ζ 2 u + 2 ζ u F{f ε (v)f ε (v u)}(ζ) ) 2 sin(2 u ζ ) u 3. (4) A(ζ) 1 + u 3, Q bε [f ε ](ζ) = Q L [f ε ](ζ) + F{f ε (v)f ε (v u)}(ζ)r(ζ, u)du R n (5) Note: this ansatz is satisfied by the Maxwellian distribution. Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

19 Numerical Results Using ε-linear: 16 x b ε (cos θ) sin θ = 8ε πθ 4 1 θ ε, Isotropic initial condition: ( f (v, ) = 1 1 e 1 ( ) ) v f(v 1 ) v 1 Solid line: solution to Landau. Dots: solution of Boltzmann with ε-linear scattering kernel. Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

20 Rate of convergence to equilibrium Landau Coulomb 1 4 ε Linear log (H H eq ) t Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21

21 Conclusions, future work Extra term in b ε - different time scale? Main issue for large scale calculation: storage Weights Ĝ(ζ, ξ) require O(N 6 ) storage. (N = 4 25 GB of memory) Flops are free - can we compute weights on the fly? work in progress (with J. Hu, I. Gamba): O(M 2 N 4 log N) algorithm requires no precomputation, but inefficient for moderate N Other anisotropic scattering potentials Multispecies/multi-energy level (with T. Magin and A. Munafo) Parallel computing: MIC vs GPU? Thank you! Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun / 21