[PPT] - A conservative spectral method for the Boltzmann equation with PowerPoint Presentation

SLIDE 1

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 2013

Work supported by NSF grants DMS-0636586 and DMS-1217254

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 1 / 21

SLIDE 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 2013 2 / 21

SLIDE 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) =

R3
S2 B(|u|, ˆ

u · σ)(f (v′

∗)f (v′) − f (v∗)f (v))dσdv∗

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 3 / 21

SLIDE 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, 00), Ibragimov-Rjasanow

(02), Pareschi-Russo (00), Mouhot-Pareschi (04), Pareschi-Russo-Toscani (00) (Landau), Pareschi-Toscani-Villani (03) (Grazing collisions consistency)

◮ Weak form: Gamba-Tharkabhushanam (09, 10) - conservation,

inelastic

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 4 / 21

SLIDE 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)

=

R3 F{f (v)f (v − u)}(ζ)G(ζ, u)du

=

u∈R3 G(u, ζ)

1 ( √ 2π)3

ξ∈R3

ˆ f (ζ − ξ)ˆ f (ξ)e−iξ·udξdu = 1 ( √ 2π)3

ξ∈R3

ˆ f (ζ − ξ)ˆ f (ξ) ˆ G(ξ, ζ)dξ

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 5 / 21

SLIDE 6

Spectral formulation of collision operator

The convolution weights G(ζ, u), ˆ G(ξ, ζ) are given by G(ζ, u) =

σ∈S2 B(|u|, cos θ)(e−i ζ

2 ·(u′−u) − 1)dσ.

ˆ G(ξ, ζ) =

u∈R3 e−iξ·u
σ∈S2 B(|u|, cos θ)(e−i ζ

2 ·(u′−u) − 1)dσdu.

For B = |u|λ/4π (isotropic), this can be reduced to ˆ G(ξ, ζ) = ∞ rλ+2[sinc(r|ζ| 2 )sinc(r|ξ − ζ 2|) − sinc(r|ξ|)]dr no time dependence

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 6 / 21

SLIDE 7

Difficulties in simulating the Boltzmann equation

Q(f , f )(v) =

R3
S2 B(|u|, ˆ

u · σ)(f (v′

∗)f (v′) − f (v∗)f (v))dσdv∗

Issues: Dimensionality: requires O(N6) operations for a single evaluation of weighted convolution

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 7 / 21

SLIDE 8

Computation is embarassingly parallel!

Test problem: 1280 spatial points, 243 velocity mesh points Time listed: wall time using Stampede (NSF XSEDE resource) for one timestep (∼244 billion operations) nodes cores time (s) 1 16 456.313 2 32 235.315 4 64 120.762 8 128 61.345 16 256 30.943 32 512 15.252 64 1024 7.813 128 2048 4.042 [H. submitted 2013, also arXiv]

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 8 / 21

SLIDE 9

Sudden Heating problem

v1/v0 g(v1)

2

2 0.1 0.2 0.3 0.4 0.5 x1/x0=0.1 x1/x0=0.3 x1/x0=0.5 x1/x0=0.9 t/t0=0.5

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 9 / 21

SLIDE 10

Difficulties in simulating the Boltzmann equation

Q(f , f )(v) =

R3
S2 B(|u|, ˆ

u · σ)(f (v′

∗)f (v′) − f (v∗)f (v))dσdv∗

Issues: Dimensionality: requires O(N6) operations for a single evaluation of weighted convolution Conservation: collision invariants need to be preserved.

v∈R3 Q(f , f )

  1 v |v|2   = 0. Solve the constrained minimization problem in O(N3) (Gamba, Tharkabhushanam)

min 1

2˜ Q − Q2

2 | CQ = 0

Jeff Haack (UT Austin)

Conservative Spectral Boltzmann 7 Jun 2013 10 / 21

SLIDE 11

Anisotropic scattering

Potentials interactions (e.g. Coulomb) include an angular component. Write B = |u|λb(cos θ). In this case, the untransformed weight G(u, ζ) is given by G(u, ζ) = |u|λ

Sd−1 b(ˆ

u · σ)(ei ζ

2 ·ue−i ζ|u| 2 ·σ − 1)dσ

= 2π|u|λ

π b(cos θ) sin θ

ei (1−cos θ)ζ

2

·uJ0

|u| sin θ|ζ⊥| 2

− 1
dθ,

ζ⊥ = ζ − (ζ · u/|u|)u/|u|.

G(ζ, ξ) = 4π2

L rλ+2 π π b(cos θ) sin θ sin φJ0

r|ξ⊥| sin φ
×
cos
r(ξ − ζ

2(1 − cos θ)) · ζ |ζ| cos φ

J0

1 2r|ζ| sin φ sin θ

− cos
rξ · ζ

|ζ| cos φ dθdφdr

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 11 / 21

SLIDE 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). QL = ∇v ·

|u|λ+2(δij − uiuj

|u|2 )(f (v∗)∇vf (v) − f (v)∇vf (v∗))

dv∗

Similar weak formulation gives GL(u, ζ) = |u|λ 4i(ζ · u) − |u|2|ζ⊥|2 Pareschi, Toscani, and Villani (2003): 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 2013 12 / 21

SLIDE 13

Approximating Boltzmann by Landau

Using Screened Coulomb: B = |u|−3 C sin4(θ/2)1θ≥ε, ε ∼ r0/λ3

D.

Isotropic initial condition: f (v, 0) = 1 100e

−10

|v|−1.5

1.5

2

. ε = 10−4, N = 16

−1 −0.5 0.5 1 2 4 6 8 10 x 10

−3

v1 f(v1)

Solid lines: Landau solution. Dashed lines: Boltzmann solution. t = 0, 1, 2, 5, 10.

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 13 / 21

SLIDE 14

Scattering kernel

Make the following assumptions on the scattering kernel. Let ε > 0 be the small parameter associated with the grazing collision limit. A family of kernels bε are grazing if (Villani, Bobylev): Λε = 2π π

0 bε(cos θ) sin2(θ/2) sin θdθ → Λ0 < ∞

∀θ0 > 0, bε(cos θ) → 0 uniformly on θ ≥ θ0 This corresponds to π

0 bε(cos θ) sin θ θkdθ → 0,

k > 2 Some examples Coulomb: bε(cos θ) sin θ = C sin θ sin4(θ/2) log sin(ε/2)1θ≥ε ε-linear: bε(cos θ) sin θ = 8ε πθ4 1θ≥ε, Note for ε-linear: π bε(cos θ) sin θ(θ2)dθ ≈ Cε θ

π

ε = C(1 − ε

π)

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 14 / 21

SLIDE 15

Convergence of Boltzmann to Landau

Qε(ζ)

=

R3 F{fε(v)fε(v − u)}(ζ)Gε(ζ, u)du

Theorem

(H., Gamba.) Assume that fε satisfies |F{fε(v)fε(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

QL[fε](ζ) −

Qbε[fε](ζ)

≤ O
1

| log sin(ε/2)|

→ 0

as ε → 0. (2)

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 15 / 21

SLIDE 16

sketch of proof

Gε(u, ζ) = |u|λ

Sd−1 bε(cos θ)(ei ζ

2 ·ue−i ζ|u| 2 ·σ − 1)dσ

= |u|λ

π 2π bε(cos θ) sin θ ×

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

ε

cos(θ/2) sin(θ/2) i(u · ζ) − 1 2 sin(θ/2) cos(θ/2)(u · ζ)2dθ =|u|−3 4i(u · ζ) − |u|2|ζ⊥|2 − |u|−3 1 2(u · ζ)2 + 1 4|ζ⊥|2|u|2 (1 + cos ε) log sin(ε/2).

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 16 / 21

SLIDE 17

sketch of proof

Qbε[fε](ζ) =

QL[fε](ζ) +

Rn F{fε(v)fε(v − u)}(ζ)R(ζ, u)du

(3) = QL[fε](ζ) +

Rn F{fε(v)fε(v − u)}(ζ)

×

|u|−3

| log sin(ε/2)| (u · ζ)2 2 + 1 4|ζ⊥|2|u|2

(1 + cos ε)

+ |u|−3 | log sin(ε/2)|

∞

n=3

Gbε,n(ζ, u)

du.

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 17 / 21

SLIDE 18

sketch of proof

|R(ζ, u)| ≤ 2|ζ|2 |u|| log sin(ε/2)|(1 + cos ε) + 2π|ζ|3 | log sin(ε/2)|

|ζ|2

|u| + 2|ζ| |u|2 + 2 √ 2 sin(2|u||ζ|) |u|3

.

(4) All integrable at zero, and |F{fε(v)fε(v − u)}(ζ)| ≤ A(ζ) 1 + |u|3 , gives integrability at ∞

Qbε[fε](ζ) =

QL[fε](ζ) +

Rn F{fε(v)fε(v − u)}(ζ)R(ζ, u)du

(5) Note: this ansatz is satisfied by the Maxwellian distribution.

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 18 / 21

SLIDE 19

Numerical Results

Using ε-linear: bε(cos θ) sin θ = 8ε πθ4 1θ≥ε, Isotropic initial condition: f (v, 0) = 1 100e

−10

|v|−1.5

1.5

2

.

−1 −0.5 0.5 1 −2 2 4 6 8 10 12 14 16 x 10

−3

v1 f(v1)

Solid line: solution to Landau. Dots: solution of Boltzmann with ε-linear scattering kernel.

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 19 / 21

SLIDE 20

Rate of convergence to equilibrium

5 10 15 20 25 30 35 10

−5

10

−4

t log (H − Heq)

Landau Coulomb ε−Linear

Jeff Haack (UT Austin) Conservative Spectral Boltzmann 7 Jun 2013 20 / 21

SLIDE 21

Conclusions, future work

’Extra’ term in bε - different time scale? Main issue for large scale calculation: storage

◮ Weights

G(ζ, ξ) require O(N6) storage. (N = 40 → ∼250 GB of memory)

◮ ”Flops are free” - can we compute weights on the fly? ◮ work in progress (with J. Hu, I. Gamba): O(M2N4 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 2013 21 / 21