on the use of the micro-macro decomposition to design

On the use of the micro-macro decomposition to design

multiscale numerical schemes for kinetic equations

Luc Mieussens

Institut de Mathematiques de Bordeaux,Universite de Bordeaux,

France

Workshop “Computational Kinetic Transport and Hybrid Methods”,2009 (IPAM, UCLA)

collaborators: P. Degond (Toulouse), G. Dimarco (Ferrara), M. Lemou (Rennes), J.-G. Liu

(Duke)

Luc Mieussens (Bordeaux) multiscale numerical schemes KTWS1 1 / 37

Kinetic equations and asymptotic models

scale factor: ε = mean free pathmacroscopic length (Knudsen number)

when ε ≪ 1: the kinetic equation is well approximated by anasymptotic ”fluid” model

Example 1: Rarefied gas dynamics (hydrodynamic limit)

∂t f + v · ∇f =1

εQ(f )

↓ ε ≪ 1

∂tU + ∇ · F (U) + ε∇ · (D∇U) = 0

(compressible Navier-Stokes eq. for U =∫(1, v , 1

2 |v |2)f dv)

↓ ε = 0

∂tU + ∇ · F (U) = 0

(compressible Euler eq.)other limits: low Mach number, boundary driven diffusion, ...


Kinetic equations and asymptotic models

scale factor: ε = mean free pathmacroscopic length (Knudsen number)

when ε ≪ 1: the kinetic equation is well approximated by anasymptotic ”fluid” model

Example 2: Linear transport (diffusion limit)

∂t f +1

εv · ∇f =

1

ε2L(f )

↓ ε = 0

∂tρ −∇ ·(κ∇ρ

)= 0

diffusion eq for ρ = 〈f 〉


Multiscale kinetic problems

multiscale: ε = O(1) in some zones, ε ≪ 1 in others

Kinetic

shoc

kFluid

��

��

diffusionkinetic

aerodynamics (Boltzmann) linear transport

usual numerical schemes: numerical constraint ∆t and ∆x = O(ε)cannot be used when ε ≪ 1

solutions:1 domain decomposition2 extended fluid models (higher order, moments, ...)3 AP schemes

this talk: a general method for (1), (2), (3)


Outline

1 AP schemesDefinitionCounter-exampleReferences

2 AP scheme for the (linear) diffusion limitMicro-macro decompositionDiscretizationNumerical tests

3 AP scheme for the compressible asymptotics of the Boltzmann equationThe Boltzmann equationAP scheme based on the micro-macro decompositionNumerical test

4 Fluid models with localized kinetic upscalingLocalization of the deviation partNumerical tests


Outline






Asymptotic Preserving (AP) schemes.

definition: a scheme uniformly stable and accurate w.r.t ε (∆t and∆x independent of ε)

consequence: in the fluid regime (ε ≪ 1): scheme consistent with thefluid equation

requirements: special time (implicit) and space discretizations


Asymptotic Preserving (AP) schemes

example of non AP scheme for the linear transport equation:

∂t f +1

εv∂x f =

1

ε2([f ] − f )

explicit upwind scheme:

f n+1 − f n

∆t+

1

ε(v+ f n

i − f ni−1

∆x+ v−

f ni+1 − f n

i

∆x) =

1

ε2([f n

i ] − f ni )

CFL constraint: ∆t ≤ average(ε2, ε∆x)numerical error O(∆x/ε)


Linear transport: contributions of Larsen, Morel, Miller,

Adams, . . .

linear transport equation (neutron transport) and its diffusion limit

fully implicit scheme (collision and transport)

main problem: AP space discretization for the steady equation

study of different space discretizations

main result: Discontinuous Galerkin (P1) approximation is AP


Unsteady equations: contributions of Klar, Jin, . . .

main ingredients:

time splitting scheme, semi-implicit (collision)decomposition of the solution (even-odd decomposition, zero-first orderdecomposition)

contribution of A. Klar: linear transport and its diffusion limit,Boltzmann equation and the low Mach number limit

contribution of S. Jin (with Pareschi, Toscani, Russo, Caflisch): lineartransport, and some hyperbolic systems with relaxation (toy modelsfor kinetic equations)

other works: Goudon-Lafitte, Gosse-Toscani, ...


Our contribution

a new kind of AP scheme

no time splitting scheme

use of the micro-macro decomposition

can be used for diffusion and hydrodynamic limits

can be used for coupling strategies


Outline






Diffusion limit of the linear transport equation[M.Lemou-LM, SISC 2008]

linear transport equation:

∂t f +1

εv∂x f =

1

ε2([f ] − f )

where [f ] =1

2

∫ 1

−1f (v) dv

limit ε = 0: diffusion equation for ρ = [f ]

∂tρ − ∂x(κ∂xρ) = 0

collision and transport are stiff


Micro-macro decomposition.

kinetic equation:

∂t f +1

εv∂x f =

1

ε2([f ] − f )

micro-macro decomposition: f = ρ + εg , where ρ = [f ]property: [g ] = 0

decomposition of the kinetic equation:

∂tρ + ε∂tg +1

εv∂xρ + v∂xg = −

1

εg

scale separation: evolution equations for ρ and g

∂tρ + ∂x [vg ] = 0,

∂tg +1

εv∂xg = −

1

ε2g − (

1

ε∂tρ +

1

ε2v∂xρ)



kinetic equation:

∂t f +1

εv∂x f =

1

ε2([f ] − f )



∂tρ + ε∂tg +1


1

εg

∂tρ is eliminated to get the micro-macro system:

∂tρ + ∂x [vg ] = 0,

∂tg +1

εv∂xg = −

1

ε2g − (

1

ε∂tρ +

1

ε2v∂xρ)



kinetic equation:

∂t f +1

εv∂x f =

1

ε2([f ] − f )



∂tρ + ε∂tg +1


1

εg


∂tρ + ∂x [vg ] = 0,

∂tg +1

εv∂xg = −

1

ε2g +

1

ε∂x [vg ] −

1

ε2v∂xρ



kinetic equation:

∂t f +1

εv∂x f =

1

ε2([f ] − f )



∂tρ + ε∂tg +1


1

εg


∂tρ + ∂x [vg ] = 0,

∂tg +1

ε(I − [.])(v∂xg) = −

1

ε2g −

1

ε2v∂xρ

(equivalent to the original equation (no approximation))


Micro-macro decomposition : another point of view.

kinetic equation:

∂t f +1

εv∂x f =

1

ε2Lf where L = [.] − I

micro-macro decomposition: f = ρ + εg , where ρ = [f ]property: [g ] = 0 ⇔ g ⊥ N (L)

decomposed kinetic equation:

∂tρ + ε∂tg +1


1

εg

scale separation in the decomposed kinetic equation:Π := [ . ], orthogonal projection onto N (L)applying Πapplying I − Π gives the evolution eq. for g


Micro-macro decomposition and the diffusion limit.

this formulation is well suited to derive the diffusion limit:

∂tρ + ∂x [vg ] = 0,

∂tg +1

ε(I − [.])(v∂xg) = −

1

ε2g −

1

ε2v∂xρ.

2nd eq. gives: g = −v∂xρ + O(ε)

use this into 1st eq. to get:

∂tρ − ∂x

([v2]

︸︷︷︸

κ

∂xρ)

= O(ε),

⇒ for ε ≪ 1, we get the diffusion limit

simple idea: numerical discretization of the micro-macro coupledsystem


Time discretization

micro-macro coupled system:

∂tρ + ∂x [vg ] = 0,

∂tg +1

ε(I − [.])(v∂xg) = −

1

ε2g −

1

ε2v∂xρ.

time semi-implicit scheme:

ρn+1 − ρn

∆t+ ∂x

[vgn+1

]= 0,

gn+1 − gn

∆t+

1

ε(I − [.])(v∂xgn) = −

1

ε2gn+1 −

1

ε2v∂xρ

n.

diffusion limit:ρn+1 − ρn

∆t− ∂xκ∂xρ

n = 0


Space discretization.

staggered grid:ρi

gi− 1

2gi+ 1

2

ρn+1i − ρn

i

∆t+ ∂x

[vgn+1

]

i= 0,

gn+1i+ 1

2

− gni+ 1

2

∆t+

1

ε(I − [.])v∂xg

ni+ 1

2

= −1

ε2gn+1i+ 1

2

−1

ε2v∂xρ

ni+ 1

2.



staggered grid:ρi

gi− 1

2gi+ 1

2

ρn+1i − ρn

i

∆t+ ∂x

[vgn+1

]

i= 0,

gn+1i+ 1

2

− gni+ 1

2

∆t+

1

ε(I − [.])v∂xg

ni+ 1

2

= −1

ε2gn+1i+ 1

2

−1

ε2v∂xρ

ni+ 1

2

.

upwind discretization of v∂xgi+ 12

(stability for kinetic regime)



staggered grid:ρi

gi− 1

2gi+ 1

2

ρn+1i − ρn

i

∆t+ ∂x

[vgn+1

]

i= 0,

gn+1i+ 1

2

− gni+ 1

2

∆t+

1

ε(I − [.])

(

v+gni+ 1

2

− gni− 1

2

∆x+ v−

gni+ 3

2

− gni+ 1

2

∆x

)

= −1

ε2gn+1i+ 1

2

−1

ε2v∂xρ

ni+ 1

2

.

upwind discretization of v∂xgi+ 12

(stability for kinetic regime)



staggered grid:ρi

gi− 1

2gi+ 1

2

ρn+1i − ρn

i

∆t+ ∂x

[vgn+1

]

i= 0,

gn+1i+ 1

2

− gni+ 1

2

∆t+

1

ε(I − [.])

(

v+gni+ 1

2

− gni− 1

2

∆x+ v−

gni+ 3

2

− gni+ 1

2

∆x

)

= −1

ε2gn+1i+ 1

2

−1

ε2v∂xρ

ni+ 1

2

.

central discretization of ∂x [vg ]i and v∂xρi+ 12

(for accurate approx. of the diffusion terms)



staggered grid:ρi

gi− 1

2gi+ 1

2

ρn+1i − ρn

i

∆t+

vgn+1i+ 1

2

− gn+1i− 1

2

∆x

= 0,

gn+1i+ 1

2

− gni+ 1

2

∆t+

1

ε(I − [.])

(

v+gni+ 1

2

− gni− 1

2

∆x+ v−

gni+ 3

2

− gni+ 1

2

∆x

)

= −1

ε2gn+1i+ 1

2

−1

ε2v

ρni+1 − ρn

i

∆x.

central discretization of ∂x [vg ]i and v∂xρi+ 12

(for accurate approx. of the diffusion terms)


Numerical diffusion limit.

the scheme:

ρn+1i

− ρni

∆t+

2

4v

gn+1

i+ 12

− gn+1

i− 12

∆x

3

5 = 0,

gn+1

i+ 12

− gn

i+ 12

∆t+

1

ε(I − [.])

0

@v+gn

i+ 12

− gn

i− 12

∆x+ v−

gn

i+ 32

− gn

i+ 12

∆x

1

A

= −

1

ε2gn+1

i+ 12

−

1

ε2v

ρni+1 − ρn

i

∆x

limit ε = 0: the scheme gives

ρn+1i − ρn

i

∆t− κ

ρni+1 − 2ρn

i + ρni−1

∆x2= 0,

time explicit scheme with 3-point stencil


Analysis[J.-G Liu-LM, 2009]

the scheme is uniformly stable:

‖ρn‖2 + ε2|||gn|||2 ≤ C(‖ρ0‖2 + ε2|||g0|||2

)

under the CFL constraint:

∆t ≤1

2(∆x2

4κ+ ε∆x)

small ε: ∆t ≤ ∆x2

8κ (CFL for diffusion)

large ε: ∆t ≤ 12ε∆x (CFL for convection)

the scheme is uniformly accurate:

‖ρ(tn) − ρn‖ + ε|||g(tn) − gn||| ≤ C (∆t + ∆x2 + ε∆x)


Numerical tests

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

explicitLM 25LM 200

ε = 1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

diffusionLM 25LM 200

ε = 10−8

0 0.1 0.2 0.3 0.4 0.50.2

0.3

0.4

0.5

0.6

0.7

explicitdiffusionLM 25LM 200JPT 25JPT 200K 25K 200

ε = 10−2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.2

0.3

0.4

0.5

0.6

0.7

diffusionLM 25LM 200JPT 25JPT 200K 25K 200

ε = 10−4


Numerical tests

0 1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2explicit 11000LM 20−10JPT 20−10K 20−10

0 1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2explicit 11000LM 100−50JPT 100−50K 100−50


Outline






The Boltzmann equation and its compressible limits.

Boltzmann equation:

∂t f + v · ∇x f =1

εQ(f , f )

ε = 0: f = M(U) ⇒ Euler equations:

∂tU + ∇x · F (U) = 0

where U = (ρ, ρu,E ) = 〈mf 〉, m = (1, v , 12 |v |

2)

ε ≪ 1: f = M(U) + O(ε) ⇒ compressible Navier-Stokes equations(CNS) :

∂tU + ∇x · F (U) = −ε

0∇x · σ

∇x · (σu + q)


A simple non-AP scheme.

Example: time splitting scheme of Coron-Perthame for the BGK equation

transport step: fn+ 1

2 −f n

∆t+ v · ∇x f

n = 0

relaxation step: f n+1 = e−∆t/εf n+ 12 + (1 − e−∆t/ε)M(f n+1)

(exact solution of: ∂t f = 1ε(M(f ) − f )

Property: scheme consistent with Euler equations, but not with CNSequations:

f n+1 − M(f n+1) = O(e−∆t/ε) ≪ O(ε)


AP scheme based on the micro-macro decomposition.[Bennoune-Lemou-LM, JCP 08]

micro-macro decomposition: f = M[U] + εgproperty: 〈mf 〉 = 〈mM[U]〉 = U and 〈mg〉 = 0

collision operator:

Q(f , f ) = Q(M(U),M(U)) + ε2Q(M(U), g) + ε2Q(g , g)

= 0 + εLM(U)g + ε2Q(g , g)

Boltzmann equation

∂tM(U) + v · ∇xM(U) + ε(∂tg + v · ∇xg) = LM(U)g + εQ(g , g)

property: g ⊥ N (LM(U))

define ΠM(U): orthogonal projection onto N (LM(U))

scale separation: apply ΠM(U) and I − ΠM(U)


AP scheme based on the micro-macro decomposition.

coupled system:

∂tU + ∇x · F (U) + ε∇x · 〈vmg〉 = 0,

∂tg + (I − ΠM(U))(v · ∇xg) =1

εLM(U)g + Q(g , g)

−1

ε(I − ΠM(U))(v · ∇xM(U)).

equivalent to the Boltzmann equation (no approximation!)

can be viewed as a fluid model (Euler) plus an upscaling term whichtakes kinetic effects into account


AP scheme based on the micro-macro decomposition.

fully discrete scheme

Un+1i − Un

i

∆t+

Fi+ 12(Un) − Fi− 1

2(Un)

∆x+ ε

*

vmgn+1

i+ 12

− gn+1

i− 12

∆x

+

= 0,

gn+1

i+ 12

− gn

i+ 12

∆t+ (I − Πn

i+ 12)

v+

gn

i+ 12− gn

i− 12

∆x+ v

−

gn

i+ 32− gn

i+ 12

∆x

!

=1

εLMn

i+ 12

gn+1

i+ 12

+ Q(gn

i+ 12, g

n

i+ 12) −

1

ε(I − Πn

i+ 12)(v

Mni+1 − Mn

i

∆x),

property: asymptotically equivalent (up to O(ε2)) to a schemewhich is:

consistent with CNS equations.second order accurate (in space) for the diffusive fluxes


Numerical test.

0 0,2 0,4 0,6 0,8 1x

-5

-2,5

0

2,5

5

7,5

10

q

APSiSeNS

0 0,2 0,4 0,6 0,8 1x

-7,5

-5

-2,5

0

2,5

5

7,5

10

q

Sod test case: heat flux (scaled)(ε = 2 × 10−3 (left) and ε = 2 × 10−4 (right))

∆t = 2 × 10−3.

AP and Si schemes: CNS regime is captured

Se scheme: CNS regime is not captured


Outline






Localization of the deviation part.[Degond-Liu-LM, MMS 06]

the micro-macro model (for Boltzmann):

∂tU + ∇x · F (U) + ∇x · 〈vmg〉 = 0,

∂tg + (I − ΠM(U))(v · ∇xg) = LM(U)g + Q(g , g)

− (I − ΠM(U))(v · ∇xM(U))

simple idea: localization of g

use the cutoff function h:determine “Fluid zones”: set h = 0determine “Kinetic zones”: set h = 1define “buffer zones” between (F) and (K) zones: set 0 ≤ h ≤ 1

define g = gK + gF with gK = hg and gF = (1 − h)g

note that gF is zero in (K) zones

localization: gF is set to 0 in (F) zones (approximation)


Localization of the deviation part.[Degond-Liu-LM, MMS 06]

the localized micro-macro model

∂tU + ∇x · F (U) + ∇x · 〈vmgK 〉 = 0,

∂tgK + h(I − ΠM(U))(v · ∇xgK ) = hLM(U)gK + hQ(gK , gK )

− h(I − ΠM(U))(v · ∇xM(U))

simple idea: localization of g

use the cutoff function h:determine “Fluid zones”: set h = 0determine “Kinetic zones”: set h = 1define “buffer zones” between (F) and (K) zones: set 0 ≤ h ≤ 1

define g = gK + gF with gK = hg and gF = (1 − h)g

note that gF is zero in (K) zones

localization: gF is set to 0 in (F) zones (approximation)


Fluid model with localized kinetic upscaling.

the localized micro-macro model:

∂tU + ∇x · F (U) + ∇x · 〈vmgK 〉 = 0,

∂tgK + h(I − ΠM(U))(v · ∇xgK ) = hLM(U)gK + hQ(gK , gK )

− h(I − ΠM(U))(v · ∇xM(U))

smooth transition between Euler in (F) zones to Boltzmann in (K)zones

dynamic transition is possible: h = h(t)

no domain decomposition

easy to implement


Numerical tests.[Degond-Liu-LM, MMS 06], [Degond-DiMarco-LM, JCP 07]

Plane shock wave for the BGK equation, static coupling.

−2 −1 0 1 2 3 4 5 60

1

2

3

4

5time=0.1

Kinetic(BGK)

Hydro. (Euler)

buffer

−2 −1 0 1 2 3 4 5 60

1

2

3

4

5time=0.3

−2 −1 0 1 2 3 4 5 60

1

2

3

4

5time=0.5

−2 −1 0 1 2 3 4 5 60

1

2

3

4

5time=0.7

−2 −1 0 1 2 3 4 5 60

1

2

3

4

5time=0.9

−2 −1 0 1 2 3 4 5 60

1

2

3

4

5time=1.2

Sod test (density and h function) for the BGK equation with dynamic localization.

−20 −15 −10 −5 0 5 10 15 20

1

2

3

4

5

6x 10

−6 Density Sod Test for t=0.018s

x(m)

dens

ity(K

g/m

3 )

kinetic modelmacroscopic modelmicmac

7.4 7.6 7.8 8 8.2 8.4

0.8

0.9

1

1.1

1.2

1.3

1.4x 10

−6

−15 −10 −5 0 5 10 15

0

0.2

0.4

0.6

0.8

1

1.2

x(m)

h, K

n x

10, Q

Interface position, Local Knudsen number and equilibrium parameter for t=0.018 s

transition functionKnudsen x 10Q

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1

tem

pera

ture

kinetic modelfluid model (diffusion)micro-Macro model

buff

er

Kinetic Fluidσ=100σ=1

Temperature for the radiative-heat transfer model, static localization.


Summary

the micro-macro decomposition is well suited for AP schemes andmultiscale methods

Perspectives

Boundary conditions for our AP schemes?2D (3D) AP schemesAP schemes for other asymptotics


Shi Jin’s remark.

Navier−Sokes

boundary layers

this scheme is not AP!!!

∆x < ε

∆t < ε

⇓

⇓

⇓

⇓


on the use of the micro-macro decomposition to design

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