Projective integration for nonlinear collisional kinetic equations W. Melis (K.U. Leuven), T. Rey and G. Samaey (K.U. Leuven) Laboratoire Paul Painlevé & Inria Rapsodi Université de Lille

May 17, 2017 Purple SHARK-FV

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

1 / 26

Outline of the talk 1

Introduction

2

Toward a high order, explicit, uniformly stable time integrator Projective Integration (PI) on a nutshell Projective Forward Euler Toward high order (and beyond?)

3

Application to kinetic equations On collisional kinetic equations Examples of kinetic models PI for collisional kinetic equation

4

Numerical Methods Summary Fast spectral method for the Boltzmann operator

5

Numerical simulations

6

Conclusion Thomas Rey (Lille 1)

Projective integration

May 17, 2017

2 / 26

Introduction

A hierarchy of fluid models for modeling a rarefied gas Microscopic: Newton equations for N -particles systems;

b

b b

b b b b b b b

b

(xi (t), vi (t))1,...,N

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

3 / 26

Introduction

A hierarchy of fluid models for modeling a rarefied gas Microscopic: Newton equations for N -particles systems; Mesoscopic: Kinetic equations (Boltzmann, Vlasov, . . . );

b

b b

b b b b b b b

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 −4 −2 v 0 y

b

2 4

4

2

0 vx

−2

−4

f ε (t, x, v)

(xi (t), vi (t))1,...,N N →∞

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

3 / 26

Introduction

A hierarchy of fluid models for modeling a rarefied gas Microscopic: Newton equations for N -particles systems; Mesoscopic: Kinetic equations (Boltzmann, Vlasov, . . . ); Macroscopic: Fluid dynamics equations (Euler, Navier-Stokes, . . . ).

b

b b

b b b b b b b

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 −4 −2 v 0 y

b

2 4

4

2

0 vx

−4

f ε (t, x, v)

(xi (t), vi (t))1,...,N N →∞

Thomas Rey (Lille 1)

−2

ρ(t, x), u(t, x), E(t, x) ε→0

Projective integration

May 17, 2017

3 / 26

Introduction

A hierarchy of fluid models for modeling a rarefied gas Microscopic: Newton equations for N -particles systems; Mesoscopic: Kinetic equations (Boltzmann, Vlasov, . . . ); Macroscopic: Fluid dynamics equations (Euler, Navier-Stokes, . . . ).

b

b b

b b b b b b b

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 −4 −2 v 0 y

b

2 4

4

2

0 vx

−2

−4

f ε (t, x, v)

(xi (t), vi (t))1,...,N N →∞

ρ(t, x), u(t, x), E(t, x) ε→0

Theoretical works: C. Cercignani, C. Bardos, R. DiPerna, P.-L. Lions, D. Levermore, C. Villani, F. Golse, L. Saint-Raymond; Numerical simulations: E. Tadmor, B. Perthame, P. Degond, L. Pareschi, E. Sonnendrücker, S. Jin, F. Filbet. Thomas Rey (Lille 1)

Projective integration

May 17, 2017

3 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Integration (PI) on a nutshell

Projective Integration “à la Gear and Kevrekidis” Let us consider the system of ODEs

( (1)

u0 (t) = g(u(t)),

t>0

N

u(0) = u0 ∈ R , Gf

Gs

where N is large and ∂g/∂u eigenvalues are clustered into two groups Gf , Gs ⊂ C, separated by a large gap (∼ stiffness): Gs is located in a neighborhood of the origin (slow components), and Gf lies far in the left-half plane (fast components). Because of the stiffness in g (through Gf ), the solution u is projected on a low dimensional equilibrium manifold in a very short time.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

4 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Integration (PI) on a nutshell

Projective Integration “à la Gear and Kevrekidis” Let us consider the system of ODEs

( (1)

u0 (t) = g(u(t)),

t>0

N

u(0) = u0 ∈ R , Gf

Gs

where N is large and ∂g/∂u eigenvalues are clustered into two groups Gf , Gs ⊂ C, separated by a large gap (∼ stiffness): Gs is located in a neighborhood of the origin (slow components), and Gf lies far in the left-half plane (fast components). Because of the stiffness in g (through Gf ), the solution u is projected on a low dimensional equilibrium manifold in a very short time. Formal idea: Perform a number of small time steps of an inner integrator, corresponding to the fast rate of damping of u towards the equilibrium manifold. Extrapolate forward with a large time step, corresponding to the slow manifold. The inner integrator can be explicit because its time steps δt will be chosen very small, e.g. δt ' O (min |λ| : λ ∈ Gf ) Thomas Rey (Lille 1)

Projective integration

May 17, 2017

4 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Forward Euler

Projective Forward Euler (PFE) scheme Gear, Kevrekidi, SINUM, 2003

(

u0 (t) = g(u(t)),

t>0

N

u(0) = u0 ∈ R ,

u(t)

Inner integrator. Forward Euler method with small time step δt: uk+1 = uk + δt g(uk ),

Thomas Rey (Lille 1)

Projective integration

k = 0, 1, . . . .

May 17, 2017

5 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Forward Euler

Projective Forward Euler (PFE) scheme Gear, Kevrekidi, SINUM, 2003

(

u0 (t) = g(u(t)),

t>0

N

u(0) = u0 ∈ R ,

u(t)

tn−1 Inner integrator. Forward Euler method with small time step δt: uk+1 = uk + δt g(uk ),

k = 0, 1, . . . .

Outer integrator. Let ∆t be a regular time step, given say by a hyperbolic CFL, and un be an approximation of the solution at time tn = n∆t First take K + 1 inner steps of size δt using the inner integrator, and denote by un,k the numerical solution at time tn,k = n∆t + kδt.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

5 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Forward Euler

Projective Forward Euler (PFE) scheme Gear, Kevrekidi, SINUM, 2003

(

u0 (t) = g(u(t)),

t>0

N

u(0) = u0 ∈ R ,

u(t)

tn tn−1 Inner integrator. Forward Euler method with small time step δt: uk+1 = uk + δt g(uk ),

tn+1

k = 0, 1, . . . .

Outer integrator. Let ∆t be a regular time step, given say by a hyperbolic CFL, and un be an approximation of the solution at time tn = n∆t First take K + 1 inner steps of size δt using the inner integrator, and denote by un,k the numerical solution at time tn,k = n∆t + kδt. Extrapolate in time (projective Forward Euler, PFE) to compute un+1 := un+1,0 un+1 = un,K+1 + (∆t − (K + 1)δt)

un,K+1 − un,K . δt

Iterate Thomas Rey (Lille 1)

Projective integration

May 17, 2017

5 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Forward Euler

Linear stability 

u0 (t) = λu(t),

t>0

u(0) = u0 ∈ R,

u(t)

tn tn−1 Inner integrator. Forward Euler method with small time step δt: uk+1 = (1 + λδt) uk = (1 + λδt)k+1 u0 ,

Thomas Rey (Lille 1)

Projective integration

tn+1

k = 0, 1, . . . .

May 17, 2017

6 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Forward Euler

Linear stability 

u0 (t) = λu(t),

t>0

u(0) = u0 ∈ R,

u(t)

tn tn−1 Inner integrator. Forward Euler method with small time step δt: uk+1 = (1 + λδt) uk = (1 + λδt)k+1 u0 ,

tn+1

k = 0, 1, . . . .

n

Outer integrator. ∆t is the regular time step, and u ' u(tn ): After K + 1 inner steps of size δt using the inner integrator: un,k = (1 + λδt)K+1 un Extrapolate in time (projective Forward Euler, PFE): un,K+1 − un,K , δt = ((M + 1)τ − M ) τ K un ,

un+1 = un,K+1 + M δt

where τ = 1 + λδt and M = ∆t/δt − (K + 1). Thomas Rey (Lille 1)

Projective integration

May 17, 2017

6 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Forward Euler

Linear stability (cont’ed) 

u0 (t) = λu(t),

t>0

u(0) = u0 ∈ R,

We have un+1

u(t)

tn tn−1 tn+1 = σ(τ )un where σ(τ ) = ((M + 1)τ − M ) τ K and M = ∆t/δt − (K + 1).

Theorem (Gear, Kevrekidis, 2003, SINUM) Let D(λ, r) = {z ∈ C : |z − λ| ≤ r}. Then δt δt |σ(τ )| ≤ 1 ⇔ τ ∈ D 1 − , ∆t ∆t



Thomas Rey (Lille 1)



Projective integration

∪D

 

δt 0, ∆t

1/K 

.

May 17, 2017

7 / 26

Toward a high order, explicit, uniformly stable time integrator

Projective Forward Euler

Linear stability (cont’ed) 

u0 (t) = λu(t),

t>0

u(0) = u0 ∈ R,

We have un+1

u(t)

tn tn−1 tn+1 = σ(τ )un where σ(τ ) = ((M + 1)τ − M ) τ K and M = ∆t/δt − (K + 1).

Theorem (Gear, Kevrekidis, 2003, SINUM) Let D(λ, r) = {z ∈ C : |z − λ| ≤ r}. Then δt δt |σ(τ )| ≤ 1 ⇔ τ ∈ D 1 − , ∆t ∆t





∪D

 

δt 0, ∆t

1/K 

.

Corollary. The PFE method is linearly stable if, and only if



λ∈D −

1 1 , ∆t ∆t

Thomas Rey (Lille 1)



 ∪D

−

1 1 , δt δt



δt ∆t

1/K 

Projective integration

May 17, 2017

7 / 26

Toward a high order, explicit, uniformly stable time integrator

Toward high order (and beyond?)

Projective Runge-Kutta method Higher-order projective Runge-Kutta (PRK) methods can be constructed by replacing each time derivative evaluation ks in a classical Runge-Kutta method by K + 1 steps of an inner integrator as follows: un,k+1

( s=1:

k1

2≤s≤S:

= un,k + δtg(un,k ), un,K+1 − un,K = δt

0≤k≤K

 n+c ,0 s u    s

= un,K+1 + (cs ∆t − (K + 1)δt)

  ks

=

s ,k+1 un+c s

=

s ,k s ,k un+c + δtg(un+c ), s s n+cs ,K+1 n+cs ,K us − us

Ps−1 as,l l=1

cs 0≤k≤K

kl ,

δt

un+1 = un,K+1 + (∆t − (K + 1)δt)

S X

b s ks .

s=1 S To ensure consistency, the Runge-Kutta matrix a = (as,i )S s,i=1 , weights b = (bs )s=1 , and nodes S c = (cs )s=1 satisfy the usual conditions 0 ≤ bs ≤ 1 and 0 ≤ cs ≤ 1, as well as: S X s=1 Thomas Rey (Lille 1)

bs = 1,

S−1 X

as,i = cs ,

1 ≤ s ≤ S.

i=1 Projective integration

May 17, 2017

8 / 26

Application to kinetic equations

On collisional kinetic equations

A general Boltzmann-like equation Scaled form Study of a particle distribution function f ε (t, x, v), depending on the time t > 0, space position x ∈ Ω ⊂ Rdx , dx ∈ {1, 2, 3} and particle velocity v ∈ Rdv , dv ≥ dx , solution to

 ε   ∂f + v · ∇x f ε = 1 Q(f ε ), ∂t

(2)

ε

  f ε (0, x, v) = f (x, v), in where Q is the collision operator, describing the microscopic collision dynamics between particles and ε is the Knudsen number, ration between the mean free path between collisions and the typical length scale.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

9 / 26

Application to kinetic equations

On collisional kinetic equations

A general Boltzmann-like equation Scaled form Study of a particle distribution function f ε (t, x, v), depending on the time t > 0, space position x ∈ Ω ⊂ Rdx , dx ∈ {1, 2, 3} and particle velocity v ∈ Rdv , dv ≥ dx , solution to

 ε   ∂f + v · ∇x f ε = 1 Q(f ε ), ∂t

(2)

ε

  f ε (0, x, v) = f (x, v), in where Q is the collision operator, describing the microscopic collision dynamics between particles and ε is the Knudsen number, ration between the mean free path between collisions and the typical length scale.

→ Huge phase space (up to 7-D!) ⇒ Deterministic numerical simulations very costly!

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

9 / 26

Application to kinetic equations

On collisional kinetic equations

A general Boltzmann-like equation Scaled form Study of a particle distribution function f ε (t, x, v), depending on the time t > 0, space position x ∈ Ω ⊂ Rdx , dx ∈ {1, 2, 3} and particle velocity v ∈ Rdv , dv ≥ dx , solution to

 ε   ∂f + v · ∇x f ε = 1 Q(f ε ), ∂t

(2)

ε

  f ε (0, x, v) = f (x, v), in where Q is the collision operator, describing the microscopic collision dynamics between particles and ε is the Knudsen number, ration between the mean free path between collisions and the typical length scale.

→ Huge phase space (up to 7-D!) ⇒ Deterministic numerical simulations very costly! → Stiff (possibly multi-scale), highly nonlinear problem ⇒ Impliciting almost impossible! Thomas Rey (Lille 1)

Projective integration

May 17, 2017

9 / 26

Application to kinetic equations

On collisional kinetic equations

Mathematical properties of the collision operator Conservation of mass, momentum and kinetic energy

Z

Z Q(f )(v) dv = 0,

Z Q(f )(v) v dv = 0,

R3

R3

Q(f )(v) |v|2 dv = 0;

R3

Dissipation of Boltzmann entropy

Z Q(f )(v) log(f )(v) dv ≤ 0; R3

Explicit equilibria, known as Maxwellian distribution



Q(f ) = 0

Thomas Rey (Lille 1)

⇔

f = Mρ,u,T :=

|v − u|2 ρ exp − 3/2 2T (2πT )

Projective integration

 ;

May 17, 2017

10 / 26

Application to kinetic equations

On collisional kinetic equations

Mathematical properties of the collision operator Conservation of mass, momentum and kinetic energy

Z

Z Q(f )(v) dv = 0,

Z Q(f )(v) v dv = 0,

R3

R3

Q(f )(v) |v|2 dv = 0;

R3

Dissipation of Boltzmann entropy

Z Q(f )(v) log(f )(v) dv ≤ 0; R3

Explicit equilibria, known as Maxwellian distribution



|v − u|2 ρ exp − 3/2 2T (2πT ) order fluid limit ε → 0 given by the compressible Euler system Q(f ) = 0

0th

⇔

f = Mρ,u,T :=

 ;

 ∂t ρ + divx (ρ u) = 0,    

∂t (ρ u) + divx (ρ u ⊗ u + ρT I) = 0R3 ,

    Thomas Rey (Lille 1)

∂t E + divx (u (E + ρT )) = 0. Projective integration

May 17, 2017

10 / 26

Application to kinetic equations

Examples of kinetic models

The Boltzmann equation It describes the non equilibrium behavior of a diluted gas of solid particles, interacting only via binary elastic collisions

Applications Microscale flow in MEMS, space shuttle atmospheric re-entry, . . .

Boltzmann collision operator Z QB (f )(v) =

f∗0 f 0 − f∗ f B(|v − v∗ |, cos θ) dσ dv∗ ,





R3 ×S2

where B is the collision kernel, cos θ := (v − v∗ ) · σ and v0 =

Thomas Rey (Lille 1)

|v − v∗ | v + v∗ + σ, 2 2

Projective integration

v∗0 =

|v − v∗ | v + v∗ − σ. 2 2

May 17, 2017

11 / 26

Application to kinetic equations

Examples of kinetic models

The Boltzmann equation It describes the non equilibrium behavior of a diluted gas of solid particles, interacting only via binary elastic collisions

Applications Microscale flow in MEMS, space shuttle atmospheric re-entry, . . .

Boltzmann collision operator v∗′

Z

v

QB (f )(v) =

f∗0 f 0 − f∗ f B(|v − v∗ |, cos θ) dσ dv∗ ,





R3 ×S2

σ

where B is the collision kernel, cos θ := (v − v∗ ) · σ and v∗

v′

v0 =

Thomas Rey (Lille 1)

|v − v∗ | v + v∗ + σ, 2 2

Projective integration

v∗0 =

|v − v∗ | v + v∗ − σ. 2 2

May 17, 2017

11 / 26

Application to kinetic equations

Examples of kinetic models

The BGK equation The BGK1 equation replaces the quadratic Boltzmann operator by a nonlinear relaxation operator which mimics its main features.

Applications Same as before, but the simpler structure of the operator allows for easier computations (with a cost in accuracy)

BGK operator 



QBGK (f )(v) = ν(rho) Mρf ,uf ,Tf (v) − f (v) , where

Z (ρf , uf , Tf ) =

f (t, x, v) ϕ(v) dv Rd

for ϕ(v) = (1, v, |v − uf |2 ) are the mass, velocity and local temperature of f and Mρ,u,T the associated Maxwellian distribution. 1 Bhatnagar,

Gross, Krook, Phys. Rev. (1954)

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

12 / 26

Application to kinetic equations

Examples of kinetic models

Riemann problem (Sod’s tube) 1Dx × 2Dv BGK vs. Boltzmann

Density ρ

Temperature T

1

Velocity ux

7.5

2

BGK 800 cells Boltzmann 800 cells

BGK 800 cells Boltzmann 800 cells

0.9

0.8

BGK 800 cells Boltzmann 800 cells

7

1.8

6.5

1.6

6

1.4

0.7 1.2

5.5 0.6

1 5 0.5

0.8 4.5

0.4

0.6 4 0.4

0.3

3.5 0.2

0.2

3 0

0.1

2.5 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

7.5

0

BGK 800 cells Boltzmann 800 cells

BGK 800 cells Boltzmann 800 cells

0.9

0.8

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.8 BGK 800 cells Boltzmann 800 cells

7

1.6

6.5

1.4

6

1.2

0.7 5.5

1

0.6 5 0.8 0.5 4.5 0.6 0.4 4 0.4 0.3

3.5 0.2

0.2

3 0

0.1

2.5 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

BGK (red) and Boltzmann (blue) solutions for ε = 10−2 (top) and ε = 10−4 , at t = 0.15 with 800 spatial cells and 642 velocity cells Thomas Rey (Lille 1)

Projective integration

May 17, 2017

13 / 26

Application to kinetic equations

PI for collisional kinetic equation

Kinetic approximation of conservation laws Let f ε ∈ L1x,v ((1 + v) dv) solution to the kinetic equation ∂t f ε + v∂x f ε =

(3) where

Z

ε

1 (R[uε ] − f ε ) ε

Z

R[u ](v)(1, v) dv = R

ε

ε



f (v) dv, g(u ) . R

Then, when ε → 0, uε converges toward u, solution to the scalar conservation law (4)

∂t u + ∂x g(u) = 0

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

14 / 26

Application to kinetic equations

PI for collisional kinetic equation

Kinetic approximation of conservation laws Let f ε ∈ L1x,v ((1 + v) dv) solution to the kinetic equation ∂t f ε + v∂x f ε =

(3) where

Z

ε

1 (R[uε ] − f ε ) ε

Z

R[u ](v)(1, v) dv = R

ε

ε



f (v) dv, g(u ) . R

Then, when ε → 0, uε converges toward u, solution to the scalar conservation law (4) ∂t u + ∂x g(u) = 0 Discretizing (3) in v on a uniform grid and in x with upwind fluxes, one can prove Theorem (Lafitte, Leijon, Melis, Samaey, 2012-2014) Choosing the parameters of the PFE scheme as K = 2, δt = ε and ∆t as the hyperbolic CFL coming from (4) provides a ε-uniformly stable time integrator for (3), whose limit is a stable approximation to (4). It is also consistent in the linear case.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

14 / 26

Application to kinetic equations

PI for collisional kinetic equation

Kinetic approximation of conservation laws Let f ε ∈ L1x,v ((1 + v) dv) solution to the kinetic equation ∂t f ε + v∂x f ε =

(3) where

Z

1 (R[uε ] − f ε ) ε

Z

ε

R[u ](v)(1, v) dv = R

ε

ε



f (v) dv, g(u ) . R

Then, when ε → 0, uε converges toward u, solution to the scalar conservation law (4) ∂t u + ∂x g(u) = 0 Discretizing (3) in v on a uniform grid and in x with upwind fluxes, one can prove Theorem (Lafitte, Leijon, Melis, Samaey, 2012-2014) Choosing the parameters of the PFE scheme as K = 2, δt = ε and ∆t as the hyperbolic CFL coming from (4) provides a ε-uniformly stable time integrator for (3), whose limit is a stable approximation to (4). It is also consistent in the linear case. Proof. Compute the slow and fast eigenvalue branches: λs = −λs1 ε + iµs (1 + ε2 ) + O(ε3 ),

λf = −

1 − λf1 ε − iµf (1 + ε2 ) + O(ε3 ) ε

then use the stability criterion of the PFE method. Thomas Rey (Lille 1)

Projective integration

May 17, 2017

14 / 26

Application to kinetic equations

PI for collisional kinetic equation

Spectrum of the linearized BGK and Boltzmann operators And now, for something (slightly) different

Denoting by M the global Maxwellian distribution M1,0,1 , one can define the linearized BGK and Boltzmann operators as LM g := M−1 (Q(M, g) + Q(g, M)) = KM g − ν(v) g where KM is a compact operator on L2v M−1 dv and ν is bounded by below. Going to Fourier in space, one can then define the linearized Boltzmann equation by




(5)

∂t g =

Thomas Rey (Lille 1)

1 KM g − (ν(v)/ε + i εγ · v) g. ε

Projective integration

May 17, 2017

15 / 26

Application to kinetic equations

PI for collisional kinetic equation

Spectrum of the linearized BGK and Boltzmann operators And now, for something (slightly) different

Denoting by M the global Maxwellian distribution M1,0,1 , one can define the linearized BGK and Boltzmann operators as LM g := M−1 (Q(M, g) + Q(g, M)) = KM g − ν(v) g where KM is a compact operator on L2v M−1 dv and ν is bounded by below. Going to Fourier in space, one can then define the linearized Boltzmann equation by




(5)

∂t g =

1 KM g − (ν(v)/ε + i εγ · v) g. ε

Theorem (Grad ’56, McLennan ’65, Nicolaenko ’71, Ellis-Pinsky ’75) The spectrum of the RHS of (5) is composed of fast modes: Eigenvalues located at a distance at least 1/ε on the left of the imaginary axis; slow modes: if |ε|  1, exactly Dv + 2 eigenvalues branches given by (j)

(j)

λ(j) (|γ|) := i λ1 ε|γ| − λ2 ε2 |γ|2 + O ε3 |γ|3 ,

Thomas Rey (Lille 1)

Projective integration




May 17, 2017

15 / 26

Application to kinetic equations

PI for collisional kinetic equation

Spectrum of the linearized BGK and Boltzmann operators And now, for something (slightly) different

Denoting by M the global Maxwellian distribution M1,0,1 , one can define the linearized BGK and Boltzmann operators as LM g := M−1 (Q(M, g) + Q(g, M)) = KM g − ν(v) g where KM is a compact operator on L2v M−1 dv and ν is bounded by below. Going to Fourier in space, one can then define the linearized Boltzmann equation by




(5)

∂t g =

1 KM g − (ν(v)/ε + i εγ · v) g. ε

Theorem (Grad ’56, McLennan ’65, Nicolaenko ’71, Ellis-Pinsky ’75) The spectrum of the RHS of (5) is composed of fast modes: Eigenvalues located at a distance at least 1/ε on the left of the imaginary axis; slow modes: if |ε|  1, exactly Dv + 2 eigenvalues branches given by (j)

(j)

λ(j) (|γ|) := i λ1 ε|γ| − λ2 ε2 |γ|2 + O ε3 |γ|3 ,




In the Boltzmann case, an essential spectrum also exists... Thomas Rey (Lille 1)

Projective integration

May 17, 2017

15 / 26

Application to kinetic equations

Thomas Rey (Lille 1)

PI for collisional kinetic equation

Projective integration

May 17, 2017

15 / 26

Application to kinetic equations

PI for collisional kinetic equation

PI for the BGK and Boltzmann equations The spectrum of the linearized BGK operator is composed of Eigenvalues located at a distance at least 1/ε on the left of the imaginary axis; If |ε|  1, Dv + 2 eigenvalues branches given by (j)

(j)

λ(j) (|γ|) := i λ1 ε|γ| − λ2 ε2 |γ|2 + O ε3 |γ|3 ,




Fast (exponential?) rate of damping of the solution to the full BGK equation toward Maxwellian distribution ⇒ Linear regime ⇒ Taking the same parameters for the PFE scheme as before K = 2, δt = ε and ∆t as the hyperbolic CFL coming from the compressible Euler dynamics will give an ε-stable, uniformly accurate, explicit time integrator for the BGK equation!

2 Gear,

Kevrekidis, SINUM 2004, Melis, Samaey, preprint 2016

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

16 / 26

Application to kinetic equations

PI for collisional kinetic equation

PI for the BGK and Boltzmann equations The spectrum of the linearized BGK operator is composed of Eigenvalues located at a distance at least 1/ε on the left of the imaginary axis; If |ε|  1, Dv + 2 eigenvalues branches given by (j)

(j)

λ(j) (|γ|) := i λ1 ε|γ| − λ2 ε2 |γ|2 + O ε3 |γ|3 ,




Fast (exponential?) rate of damping of the solution to the full BGK equation toward Maxwellian distribution ⇒ Linear regime ⇒ Taking the same parameters for the PFE scheme as before K = 2, δt = ε and ∆t as the hyperbolic CFL coming from the compressible Euler dynamics will give an ε-stable, uniformly accurate, explicit time integrator for the BGK equation! In the Boltzmann case, an essential spectrum also exists... Need to use Telescopic Projective Integration2 , which brings a log(1/ε) dependency on δt. But this is another story ;-)

2 Gear,

Kevrekidis, SINUM 2004, Melis, Samaey, preprint 2016

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

16 / 26

Numerical Methods

Summary

Summary of the numerical solvers Numerically solving the kinetic equation ∂t f + v · ∇ x f =

1 Q(f ) ε

Introduce a Cartesian grid V of RDv by V = {vk = k∆v + a, k ∈ K} and denote the discrete collision invariants on V by mk = (1, vk , 21 |vk |2 ). Replace the continuous distribution function f by a N -vector fK (x, t), where each component is assumed to be an approximation of f at location vk : fk (x, t) ≈ f (x, vk , t). The fluid quantities are then obtained from fk : U (x, t) =

X

mk fk (x, t) ∆v.

k

The discrete velocity model becomes a set of N equations for fk ∂t fk + vk · ∇x fk = Q(fk ), where the term Q(fk ) couples all the equations. Free transport term divx (vk fk ) computed with WENO reconstruction. PRK time stepping. Thomas Rey (Lille 1)

Projective integration

May 17, 2017

17 / 26

Numerical Methods

Fast spectral method for the Boltzmann operator

Spectral discretization of Boltzmann collision operator Truncation of the Boltzmann operator (assume now that f = f (v) only): If the distribution function √ f have compact support on B0 (R), then supp(Q(f, f )(v)) ⊂ B0 ( 2R). Thus, to write a spectral approximation which √ avoids aliasing, it is sufficient that f (v) is restricted to [−T, T ]Dv with T ≥ (2 + 2)R. Assuming f (v) = 0 on [−T, T ]Dv \ B0 (R), we extend f (v) to a periodic function on the set [−T, T ]3 . √ The choice T = (3 + 2)R/2 guarantees the absence of intersection between periods where f is different from zero.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

18 / 26

Numerical Methods

Fast spectral method for the Boltzmann operator

Spectral discretization of Boltzmann collision operator Truncation of the Boltzmann operator (assume now that f = f (v) only): If the distribution function √ f have compact support on B0 (R), then supp(Q(f, f )(v)) ⊂ B0 ( 2R). Thus, to write a spectral approximation which √ avoids aliasing, it is sufficient that f (v) is restricted to [−T, T ]Dv with T ≥ (2 + 2)R. Assuming f (v) = 0 on [−T, T ]Dv \ B0 (R), we extend f (v) to a periodic function on the set [−T, T ]3 . √ The choice T = (3 + 2)R/2 guarantees the absence of intersection between periods where f is different from zero. Fourier representation of the collision operator: Let us take T = π and hence R = λπ with λ = 2/(3 +

√

2).

The distribution function is represented as the truncated Fourier series fN (v) =

N X k=−N

Thomas Rey (Lille 1)

fˆk eik·v ,

fˆk =

1 (2π)Dv

Projective integration

Z

f (v)e−ik·v dv.

[−π,π]Dv

May 17, 2017

18 / 26

Numerical Methods

Fast spectral method for the Boltzmann operator

Spectral discretization of Boltzmann collision operator II Z

f∗0 f 0 − f∗ f B(|v − v∗ |, cos θ) dσ dv∗ ,



QB (f )(v) =



R3 ×S2

We then obtain a spectral quadrature by projecting the Boltzmann operator on the space of trigonometric polynomials of degree ≤ N , i.e. ˆk = Q

Z

Q(fN )e−ik·v dv,

k = −N, . . . , N.

[−π,π]3

ˆ one gets By substituting the truncated Fourier series fN in Q N X

ˆk = Q

ˆ m), fˆl fˆm β(l,

k = −N, . . . , N,

l,m=−N l+m=k

ˆ m) = B(l, m) − B(m, m) are given by β(l,

Z

Z

B(l, m) = B0 (2λπ)

with q + = 12 (q + |q|ω),

|q|σ(|q|, cos θ)e−i(l·q

+

+m·q − )

dω dq.

S2

q − = 12 (q − |q|ω).

The evaluation of B(l, m) requires O(N 2 ) operations. Thomas Rey (Lille 1)

Projective integration

May 17, 2017

19 / 26

Numerical Methods

Fast spectral method for the Boltzmann operator

Fast spectral discretization In order to reduce the number of operations needed to evaluate the collision integral, the main idea is to use the so-called Carleman representation. This gives

Z

Z

˜ B(x, y)δ(x · y) [f (v + y) f (v + x) − f (v + x + y) f (v)] dx dy,

QB (f ) = R3

R3

with dv −1

˜ B(|x|, |y|) = 2

σ

p

|x|2 + |y|2 , p

|x|

!

|x|2 + |y|2

(|x|2 + |y|2 )−

dv −3 2

.

This transformation permits to get to the following new spectral quadrature formula N X

ˆk = Q

βˆF (l, m) fˆl fˆm ,

k = −N, ..., N

l,m=−N l+m=k

where βˆF (l, m) = BF (l, m) − BF (m, m) are now given by

Z

Z

˜ B(x, y) δ(x · y) ei(l·x+m·y) dx dy.

BF (l, m) = B0 (R) Thomas Rey (Lille 1)

B0 (R) Projective integration

May 17, 2017

20 / 26

Numerical Methods

Fast spectral method for the Boltzmann operator

Fast spectral discretization II Now, we look for a convolution structure. The aim is to approximate each βˆF (l, m) by a sum βˆF (l, m) '

A X

αp (l)αp0 (m)

p=1

This gives a sum of A discrete convolutions and so the algorithm can be computed in O(A N log2 N ) operations by means of standard FFT techniques.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

21 / 26

Numerical Methods

Fast spectral method for the Boltzmann operator

Fast spectral discretization II Now, we look for a convolution structure. The aim is to approximate each βˆF (l, m) by a sum βˆF (l, m) '

A X

αp (l)αp0 (m)

p=1

This gives a sum of A discrete convolutions and so the algorithm can be computed in O(A N log2 N ) operations by means of standard FFT techniques. An example, the two dimensional case: Make the decoupling assumption ˜ B(x, y) = a(|x|) b(|y|); ˜ is constant (2D Maxwellian molecules, 3D hard spheres). satisfied if e.g. B This gives

Z BF (l, m) =

π

φ2R (l · eθ ) φ2R (m · eθ+π/2 ) dθ,

φ2R (s) = 2 R sinc(Rs).

0

A regular discretization of M equally spaced points gives BF (l, m) =

M −1 π X αp (l)αp0 (m), M

αp (l) = φ2R (l · eθp ), αp0 (m) = φ2R (m · eθp +π/2 )

p=0

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

21 / 26

Numerical simulations

1Dx − 1Dv BGK Sod shock tube problem, PRK4 time integrator, WENO 3 in x

First moments of the solution to the BGK equation with ν = 1 (left) and ν = ρ (right) Density 1

0.8

0.8 ρ(x, t)

ρ(x, t)

Density 1

0.6 0.4 0.2

0.6 0.4 0.2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x Velocity

0.8

1

0.8

1

Velocity

0.8

0.8

0.6

0.6

v ¯(x, t)

v ¯(x, t)

0.6 x

0.4 0.2

0.4 0.2

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6 x

x

0.8

Thomas Rey (Lille 0.6 1)

x, t)

x, t)

∆t = 0.4∆x, ∆x = 0.01, Temperature Nv = 80, K = 2 and δt = ε, for Temperature ε = 10−1 (blue dots), 10−2 −5 1 1 (purple dots), and 10 (green dots). Red line: hydrodynamic limit ε → 0 0.8

Projective integration 0.6

May 17, 2017

22 / 26

Numerical simulations

v ¯(x, t)

v ¯(x, t)

0.6 0.4

1Dx − 1Dv BGK 0.2

0.6 0.4 0.2

0 Sod shock tube problem, PRK4 time integrator, WENO0 3 in x 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

First moments of the solutionxto the BGK equation with ν = 1x (left) and ν = ρ (right) Temperature 1

0.8

0.8

T (x, t)

T (x, t)

Temperature 1

0.6 0.4

0.6 0.4

0.2

0.2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x Heat flux

0.05

0

−0.05

0

0.2

0.4

0.8

1

0.6

0.8

1

Heat flux

0.1

q(x, t)

q(x, t)

0.1

0.6 x

0.8

1

0.05

0

−0.05

x

0

0.2

0.4

0.6 x

∆t = 0.4∆x, ∆x = 0.01, Nv = 80, K = 2 and δt = ε, for ε = 10−1 (blue dots), 10−2 (purple dots), and 10−5 (green dots). Red line: hydrodynamic limit ε → 0 Thomas Rey (Lille 1)

Projective integration

May 17, 2017

22 / 26

Numerical simulations

1Dx − 2Dv BGK vs. Boltzmann Sod shock tube problem, PRK4 time integrator, WENO 2 in x, fast spectral in v

First moments of BGK equation with ν = 1 (blue), ν = ρ (green) and Boltzmann (red) Density 1

0.8

0.8 ρ(x, t)

ρ(x, t)

Density 1

0.6 0.4 0.2

0.6 0.4 0.2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x Velocity

0.8

1

0.8

1

Velocity

1

1

0.8

0.8

0.6

0.6

v ¯(x, t)

v ¯(x, t)

0.6 x

0.4 0.2

0.4 0.2

0

0 0

0.2

0.4

0.6

0.8

1

0

x

0.2

0.4

0.6 x

0.8

Thomas Rey (Lille 0.6 1)

x, t)

x, t)

∆t = 0.4∆x, ∆x = 0.01, Temperature Nv = 322 , K = 2 and δt = ε, forTemperature ε = 10−2 (left), and 10−5 1 1 (right). 0.8

Projective integration 0.6

May 17, 2017

23 / 26

0.4

v ¯(x, t)

v ¯(x, t)

Numerical simulations

0.6

0.6

1Dx − 2Dv BGK vs. Boltzmann 0.2

0.4 0.2

0 Sod shock tube problem, PRK4 time integrator, WENO0 2 in x, fast spectral in v 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x x First moments of BGK equation with ν = 1 (blue), ν = ρ (green) and Boltzmann (red) Temperature 1

0.8

0.8

T (x, t)

T (x, t)

Temperature 1

0.6 0.4

0.6 0.4

0.2

0.2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x Heat flux

0.08

0.08

0.06

0.06

0.04 0.02

0 −0.02

0.4

0.8

1

0.02

−0.02

0.2

1

0.04

0 0

0.8

0.6

Heat flux

0.1

q(x, t)

q(x, t)

0.1

0.6 x

0.8

1

x

0

0.2

0.4

0.6 x

∆t = 0.4∆x, ∆x = 0.01, Nv = 322 , K = 2 and δt = ε, for ε = 10−2 (left), and 10−5 (right). Thomas Rey (Lille 1)

Projective integration

May 17, 2017

23 / 26

Numerical simulations

2Dx − 2Dv BGK Shock-Bubble interaction, PRK4 time integrator, WENO 2 in x, ε = 10−5 , ν = 1 Density (t = 0)

2.5

ρ(x, 0)

2

1.5 1 1 −2

0 −1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

−1

2

2.5

3

−1

x

y

Temperature (t = 0)

2.5

T (x, 0)

2 1.5 1 1 0.5 −2

0 −1.5

−1

−0.5

0

0.5

1

1.5

x

y

2

∆t = 0.4∆x, Nx = 200 × 25, Nv = 32 , K = 2 and δt = ε. Thomas Rey (Lille 1)

Projective integration

May 17, 2017

24 / 26

Numerical simulations

2Dx − 2Dv BGK Shock-Bubble interaction, PRK4 time integrator, WENO 2 in x, ε = 10−5 , ν = 1 Density (t = 0.8)

4.5 4 ρ(x, 0.8)

3.5 3 2.5 2 1

1.5 1 −2

0 −1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

−1

2

2.5

3

−1

x

y

Temperature (t = 0.8)

T (x, 0.8)

2.5

2

1.5 1 1 −2

0 −1.5

−1

−0.5

0

0.5

1

1.5

x

y

2

∆t = 0.4∆x, Nx = 200 × 25, Nv = 32 , K = 2 and δt = ε. Thomas Rey (Lille 1)

Projective integration

May 17, 2017

24 / 26

Numerical simulations

2Dx − 2Dv Boltzmann Double Sod shock, TPRK4 time integrator, WENO 2 in x, ε = 5.10−5 , ν = 1

Boltzmann equation in 2D/2D (level-2 TPRK4 + WENO2) Velocity (x)

Density 1 0.4

0.8

0.4 0.8

0.4

0.2 0.6

0

y

y

0.2

−0.2

0.4

−0.2

−0.4

0.2

−0.4

−0.4 −0.2

0

0.2

0

0

−0.4 −0.8 −0.4 −0.2

0.4

x Energy

0

0.2

0.4

x

Pressure

1

1

∆t =0.40.4∆x, Nx = 642 , Nv = 322 , K = 3 and δt = ε. 0.4 0.8 0.2 Thomas Rey (Lille 1)

0.8 0.2

Projective integration

May 17, 2017

25 / 26

Numerical simulations

0.4

−0.2

2Dx − 2Dv Boltzmann

0.4

−0.2

0.2

0.2

−0.4Sod shock, TPRK4 time integrator, WENO−0.4 Double 2 in x, ε = 5.10−5 , ν = 1 −0.4 −0.2

0

0.2

−0.4 −0.2

0.4

x Temperature

0.2

0.4

Mach number 2.5

0.4

0

1.5

−0.2

y

2

0.2 y

0 x

0.4

1

0.2

0.8 0.6

0

1

−0.2

0.5

−0.4

0.4 0.2

−0.4 −0.4 −0.2

0

0.2

−0.4 −0.2

0.4

x

0

0.2

0.4

0

x

∆t = 0.4∆x, Nx = 642 , Nv = 322 , K = 3 and δt = ε.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

25 / 26

Conclusion

Conclusion We have built and implemented a deterministic, high order, explicit and asymptotic preserving solvers for nonlinear kinetic equations; The method is very easy to implement, since its basic building block is the forward Euler scheme; Need to know some spectral properties of the equation.

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

26 / 26

Conclusion

Conclusion We have built and implemented a deterministic, high order, explicit and asymptotic preserving solvers for nonlinear kinetic equations; The method is very easy to implement, since its basic building block is the forward Euler scheme; Need to know some spectral properties of the equation.

TODO What if the spectrum doesn’t separate?

TODO What about consistency?

TODO What about uniform accuracy?

Thomas Rey (Lille 1)

Projective integration

May 17, 2017

26 / 26

Conclusion

Conclusion We have built and implemented a deterministic, high order, explicit and asymptotic preserving solvers for nonlinear kinetic equations; The method is very easy to implement, since its basic building block is the forward Euler scheme; Need to know some spectral properties of the equation.

TODO What if the spectrum doesn’t separate?

TODO What about consistency?

TODO What about uniform accuracy?

Thanks a lot for your attention! Thomas Rey (Lille 1)

Projective integration

May 17, 2017

26 / 26