implicit schemes for the equation of the bgk model · boltzmann equation for the evolution of a...

The BGK modelNumerical schemes for BGK models

MiMe numerical schemes for BGK modelsCompressible NS asymptotics

Implicit schemes for the equation of the BGKmodel

Sandra Pieraccini, Gabriella Puppo

Dipartimento di Scienze MatematichePolitecnico di Torino

http://calvino.polito.it/~ puppo

[email protected]

International Conference on Hyperbolic Problems:Theory, Numerics, Applications

Padova, June 25-29, 2012

Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models



Motivation for BGK model

The BGK model (Bhatnagar-Gross-Krook ’54) approximatesBoltzmann equation for the evolution of a rarefied gas for smalland moderate Knudsen numbers:

Kn =mean free path

characteristic length of the problem

Lately, interest in this model has increased because:

Several desirable properties have been shown to hold for theBGK model and its variants, such as BGK-ES, (Perthame etal. from 1989 on)






Kn =mean free path



The BGK model has been extended to include more generalfluids and can now be applied to the flow of a polytropic gas(Mieussens) and to mixtures of reacting gases (Aoki et al.)






Kn =mean free path



New applications of kinetic models have appeared. Forinstance, fluid flow in nanostructures can be described by theBGK model, since it occures at moderate Knudsen numbers




Outline

The main topics of the talk

The BGK equation and its properties

Numerical difficulties

Microscopically Implicit, Macroscopically Explicit (MiMe)schemes

Numerical examples

Asymptotic properties of MiMe schemes




Outline

The BGK equation and its properties

Numerical difficulties

Microscopically Implicit, Macroscopically Explicit (MiMe)schemes

Numerical examples

Asymptotic properties of MiMe schemes




BGK model

The main variable is the mass density f of particles in the pointx ∈ Rd with velocity v ∈ RN at time t, thus f = f (x , v , t). Theevolution of f is given by:

∂f

∂t+ v · Ox f =

1

τ(fM − f ) ,

with initial condition f (x , v , 0) = f0(x , v) ≥ 0. With this notationf (x , v , t) becomes a probability density dividing by ρ(x , t).Here τ is the collision time τ ' Kn, so τ > 0 and in thehydrodynamic regime τ can be very small.




The Maxwellian

fM is the local Maxwellian function, and it is built starting fromthe macroscopic moments of f :

fM(x , v , t) =ρ(x , t)

(2πRT (x , t))N/2exp

(−||v − u(x , t)||2

2RT (x , t)

),

where ρ and u are the gas macroscopic density and velocity and Tis the temperature. They are computed from f as: ρ

ρuE

=

⟨f

1v

12 ||v ||

2

⟩ where 〈g〉 =

∫RN

g dv

E is total energy, and the temperature is: NRT/2 = E − 1/2ρu2,where N is the number of degrees of freedom in velocity




The Maxwellian

Thus the BGK equation

∂f

∂t+ v · Ox f =

1

τ(fM − f ) ,

describes the relaxation of f towards the local equilibriumMaxwellian fM .The local equilibrium is reached with a speed that is inverselyproportional to τ . Thus the system is stiff for τ << 1.




Conservation

Since ρρuE

=

⟨f

1v

12 ||v ||

2

⟩ =

⟨fM

1v

12 ||v ||

2

⟩




Conservation

As in Boltzmann equation, the first macroscopic moments of f areconserved:

∂t 〈f 〉+∇x · 〈fv〉 = 0,

∂t 〈fv〉+∇x · 〈v ⊗ vf 〉 = 0,

∂t

⟨12‖v‖

2f⟩

+∇x ·⟨

12‖v‖

2vf⟩

= 0.

Thus a numerical scheme for the BGK model must be conservative,and its numerical solution must converge to the Euler solution asKn→ 0.




Conservation

As in Boltzmann equation, the first macroscopic moments of f areconserved:

∂t 〈f 〉+∇x · 〈fv〉 = 0,

∂t 〈fv〉+∇x · 〈v ⊗ vf 〉 = 0,

∂t

⟨12‖v‖

2f⟩

+∇x ·⟨

12‖v‖

2vf⟩

= 0.

Moreover, for Kn → 0 the macroscopic solution converges to thegas dynamic solution of Euler equations.




Entropy principle

The BGK model satisfies an entropy principle:

∂t 〈f log f 〉+∇x 〈vf log f 〉 ≤ 0, ∀f ≥ 0

where equality holds if and only if f = fM . Thus the MaxwellianfM is the equilibrium solution of the system.The macroscopic entropy is:

S = 〈f log f 〉

Note that as τ → 0, entropy is conserved on smooth solutions, asfor Euler solutions.




Numerical schemes for the BGK model

The development of numerical methods for the BGK model hasstarted only recently.

Yang, Huang ’95This scheme is high order accurate in space, but only firstorder accurate in time






Aoki, Kanba, Takata ’97This is a second order scheme, designed for smooth solutions






Mieussens, ’00Second order schemes, where conservation is exactly enforced.Both explicit and implicit case are considered.

Bennoune, Lemou, Mieussens, ’08Micro-Macro decomposition






Andries, Bourgat, le Tallec, Perthame ’02A stochastic Monte Carlo scheme






Pieraccini, Puppo SISC ’06IMEX schemes for the BGK model. Non oscillatory high orderschemes in space and time. The schemes are implicit in therelaxation part.

Pieraccini, Puppo JCP ’11Microscopically Implicit Macroscopically Explicit schemes forthe BGK equation

Alaia, Puppo, JCP ’12A hybrid method for hydrodynamic and kinetic flow, Part II:Coupling of hydrodynamic and kinetic models






Russo, SantagatiLagrangian scheme

Filbet, Jin, JCP 2010A class of asymptotic-preserving schemes for kinetic equationsand related problems with stiff sources.

F. Filbet and S. Jin, JSC 2011An asymptotic preserving scheme for the ES-BGK model ofthe Boltzmann equation




Numerical difficulties of BGK models

A numerical scheme for the BGK model must satisfy severalconstraints

It must satisfy the same conservation properties of the exactmodel

It must satisfy the same asymptotic properties of the exactmodel. Therefore, it must provide a consistent discretizationof Euler equations, when Kn→ 0, and, possibly it shouldbecome a discretization of the compressible Navier Stokesequations when Kn is small.

The BGK model is stiff both in the relaxation term and in thehigh microscopic speeds.

The solution f must remain positive for all time, and shouldsatisfy an entropy condition.

It must reduce to free flow for Kn→∞




Numerical difficulties of BGK models

A numerical scheme for the BGK model must satisfy severalconstraints

It must satisfy the same conservation properties of the exactmodel

It must satisfy the same asymptotic properties of the exactmodel. Therefore, it must provide a consistent discretizationof Euler equations, when Kn→ 0, and, possibly it shouldbecome a discretization of the compressible Navier Stokesequations when Kn is small.

The BGK model is stiff both in the relaxation term and in thehigh microscopic speeds.

The solution f must remain positive for all time, and shouldsatisfy an entropy condition.

It must reduce to free flow for Kn→∞Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models



Motivation for MiMe schemes for BGK models

In the following we will attempt the construction of a schemewhich is implicit in the stiff source terms and in the fast convectivemodes, while still being explicit in the convective term on themacroscale, which determine the Maxwellian.

1 The scheme is implicit in the relaxation and in the convectiveterms

2 The computation of the Maxwellian is still carried outexplicitly. So the main non linearity of the BGK model istreated explicitly.

3 The stability condition which determines the timestep isdictated by the macroscopic modes




Structure of the implicit scheme

Let f nj ,k = f (xj , vk , t

n) and:

∆F (f n)j ,k = Fj+1/2(f nk )− Fj−1/2(f n

k )

be the convective flux difference. Then the first order discretizedequation for f will be written as:

f n+1j ,k = f n

j ,k − λvk∆F (f n+1)j ,k +∆t

τn+1j

((fM)n+1

j ,k − f n+1j ,k

)The problem is that we cannot evaluate the moments at time

level n + 1 starting from known quantities, because the momentsare not known at time tn+1.




Computation of moments

We use an explicit discretization of the moments equations:

∂t 〈f 〉+∇x · 〈fv〉 = 0,

∂t 〈fv〉+∇x · 〈v ⊗ vf 〉 = 0,

∂t

⟨12‖v‖

2f⟩

+∇x ·⟨

12‖v‖

2vf⟩

= 0.

where the fluxes 〈fv〉, 〈v ⊗ vf 〉 and⟨

12‖v‖

2vf⟩

are computed fromf n. From these equations we obtain ρn+1, un+1 and T n+1, underthe macroscopic CFL:

max (|u|+ c) ∆t ≤ h

where c is the sound speed.




Numerical macroscopic fluxes

Write the macroscopic moment equations as:

∂tu(f ) = −∂xG(f ),






∂tu(f ) = −∂xG(f ),

where u and G are

u =

〈f 〉

〈fv〉⟨12‖v‖

2f⟩

G =

〈fv〉

〈v ⊗ vf 〉⟨12‖v‖

2vf⟩






∂tu(f ) = −∂xG(f ),

Then the equation can be discretized in space as

∂tu(f ) = −1

h

(Gj+1/2(u)− Gj−1/2(u)

)where the numerical flux Gj+1/2 = G(u−j+1/2,u

+j+1/2), and u±j+1/2 are

the left and right boundary extrapolated data at the cell interfaces,obtained from the reconstruction, applied to u. As numerical flux,one can use the Lax Friedrichs flux splitting, or the HLL flux.




Time integration

We integrate in time the macroscopic equations with an explicitRunge-Kutta scheme:

u(f )n+1j = u(f )n

j − λ∑

l

bl∆Gj(u(f (l))) (1)

u(f (l))j = u(f )nj − λ

l−1∑k=1

al ,k∆Gj(u(f (k))) (2)

For the second order scheme, this requires to estimate f (2) at thenew time level tn + ∆t. This is done solving the implicit equationfor f with the implicit Euler scheme. We believe that this can begeneralized to higher order schemes, because in all cases, the RKstep is composed of first order Euler steps.




Second order integration for f

Implicit time integration can be very diffusive. For this reason, inthe second order case, we choose the Crank Nicolson scheme.

f n+1j ,k = f n

j ,k −λ

2vk

[∆F (f n+1)j ,k + ∆F (f n)j ,k

]+

∆t

2

[1

τn+1j

((fM)n+1

j ,k − f n+1j ,k

)+

1

τnj

((fM)n

j ,k − f nj ,k

)]

1 The second order space discretization for f uses a secondorder upwind formula, which is based on the evaluation oflimited slopes. This introduces non linearities in the system ofequations for f n+1.

2 To avoid non linearities, we choose the formula on which theslope is based using as predictor the previous evaluation off (2) obtained while updating the moment equations.

3 The same formula is used to compute the slopes of f n+1, sothat now the space discretization is linear in f n+1.




Second order integration for f

1 The second order space discretization for f uses a secondorder upwind formula, which is based on the evaluation oflimited slopes. This introduces non linearities in the system ofequations for f n+1.

2 To avoid non linearities, we choose the formula on which theslope is based using as predictor the previous evaluation off (2) obtained while updating the moment equations.

3 The same formula is used to compute the slopes of f n+1, sothat now the space discretization is linear in f n+1.




Convergence rate

Convergence rate on macroscopic quantities for Kn = 10−1 andKn = 10−2 in the L1 norm, on a smooth profile.




Realigning moments

The integration of moments is conservative, because the numericalfluxes are conservative. However, f n+1 does not correspond exactlyto the moments un+1, which are only predicted from the old valuesf n. In fact we can write:

un+1 = Hu(un, f n)

f n+1 = Hf (un+1, f n)

This effect becomes more important for large Knudsen numbers.So:

1 Evaluate a local Knudsen number as: Knloc = Kn/ρn+1x . If

Knloc > 0.1 in some cells, realign moments, i.e. setun+1 =

⟨f n+1φ(v)

⟩.

2 With this correction moments are corrected only in the firsttime steps for large Kn.

3 This device prevents instabilities and it accelerates theconvergence rate in the kinetic regime.




Realigning moments

1 Evaluate a local Knudsen number as: Knloc = Kn/ρn+1x . If

Knloc > 0.1 in some cells, realign moments, i.e. setun+1 =

⟨f n+1φ(v)

⟩.

2 With this correction moments are corrected only in the firsttime steps for large Kn.

3 This device prevents instabilities and it accelerates theconvergence rate in the kinetic regime.




Realignement of moments-1

MiMe1, Kn = 0.1, temperature profile without (left) and with(right) realignement for several grids




Realignement of moments-2

MiMe2, Kn = 0.1, temperature profile without (left) and with(right) realignement for several grids




Changing the order

MiMe, Kn = 10−5, density and temperature profiles with first orderMiMe scheme (cyan), second order MiMe (magenta) and thirdorder explicit (black)




Compressible Navier Stokes

To study the asymptotic properties of MiMe schemes, we considerthe simple case of 1 degree of freedom, both in space and in velocity.We write ετ instead of τ to emphasize the small parameter in thekinetic correction. The Compressible Navier Stokes equations in thiscase reduce to:

∂t

ρmE

+ ∂x

m2E

mρ (E + ρT )

= ε∂x

00

32τρT∂xT

so that the non-equilibrium correction occurs only in the energy

equation.




Compressible Navier Stokes

Thus the equations we are solving with the BGK scheme in this casereduce to:

∂tρ+ ∂x ·m = 0,

∂tm + ∂x · (2E ) = 0,

∂tE + ∂x ·⟨

1

2‖v‖2vf

⟩= 0.

and we want to study the heat flux correction resulting from ourschemes. For simplicity we consider only the first order semidiscretein time scheme, with one degree of freedom both in space and inmicroscopic velocity.




Asymptotics for semidiscrete MiMe scheme

We consider the first order in time semidiscrete MiMe scheme. Inthe first time step, we set U0 =

⟨φf 0⟩. We write Mn = M(Un),

where M(U) is the Maxwellian built with the moments U. Then thesemidiscrete in time first order scheme can be written as:

Un+1 −Un

∆t= −∂x

mn

2En⟨12v3f n

⟩

Mn+1 = M(Un+1)

f n+1 − f n

∆t= −v∂x f

n+1 +1

ετ

(Mn+1 − f n+1

).





Here, Mn and f n do not have exactly the same moments, but asε → 0, f n → Mn. Thus we decompose f in its equilibrium andkinetic part as: f n = Mn + εgn, but recalling that 〈φg〉 6= 0. Still,we can compute the first order kinetic correction starting from theequation for f , finding:

gn = − τ

∆t

(Mn −Mn−1

)− τv∂xM

n + O(ε)





Substituting the kinetic correction in the energy equation, we seethat the flux becomes:⟨

1

2v3f n

⟩=

⟨1

2v3Mn

⟩−

ετ

[⟨1

2v3

(Mn −Mn−1

)∆t

⟩+ ∂x

⟨1

2v4Mn

⟩]+ O(ε2)





Substituting the kinetic correction in the energy equation, we seethat the flux becomes:⟨

1

2v3f n

⟩=

⟨1

2v3Mn

⟩−

ετ

[∂t

⟨1

2v3Mn

⟩+ ∂x

⟨1

2v4Mn

⟩]+ O(ε2) + O(ε∆t)





Using the expressions⟨v3M

⟩= ρu3 + 3ρuT ,

⟨v4M

⟩= ρu4 +

6ρu2T + 3ρT 2 and the conservation laws at order ε0, we recover:

Un+1 −Un

∆t+∂x

m2E

mρ (E + ρT )

= ε∂x

00

32τρT∂xT

+O(ε∆t+ε2)

which is the Navier Stokes equation, corresponding to one degreeof freedom in velocity space (which gives no shear viscosity).Thus the semidiscrete first order MiMe scheme is consistent withthe correct equation as ∆t → 0.




Convergence to CNS

MiMe2. Convergence to Compressible Navier-Stokes. Left to right:Kn = 0.1, Kn = 0.05




Convergence to CNS

MiMe2. Convergence to Compressible Navier-Stokes. Left to right,Kn = 0.02 and Kn = 0.01




Stabilization and realignement

The CNS solver is explicit and it uses a CFL:

∆t = 0.9 min

(h

α,

h2

3KnTMρM

)where α = maxx(|u|+

√3T ) which can be quite penalising when

Kn is relatively high, while MiMe scheme travels with a CFL:

∆t = 0.9h

α

The improved stability region is given by realignement. Let us seehow it works...




Semidiscrete MiMe with realignement

The first order in time semidiscrete MiMe scheme with realignementcan be written as follows. Given f n:

Un = 〈φf n〉

Un+1 − Un

∆t= −∂x

mn

2En⟨12v3f n

⟩

Mn+1 = M(Un+1)

f n+1 − f n

∆t= −v∂x f

n+1 +1

ετ

(Mn+1 − f n+1

).





Again, Mn and f n do not have exactly the same moments, but asε → 0, f n → Mn. Thus we decompose f as: f n = Mn + εgn,but recalling that 〈φg〉 6= 0. Now the macroscopic equation can bewritten as:

Un+1 −Un

∆t= −〈vφ∂x f

n〉+Un −Un

∆t

which has the same asymptotics than the semidiscrete MiMescheme, because the added term satisfies:

Un+1 − Un+1

∆t=

ετ

ετ + ∆t∂x

⟨vφ(f n+1 − f n

)⟩= O

(ε∆t

ετ + ∆t

).





On the other hand, if we consider the evolution equation for U, wehave:

Un+1 − Un

∆t= −

[ετ

ετ + ∆t∂x

⟨vφf n+1

⟩+

∆t

ετ + ∆t∂x 〈vφf n〉

].

In other words, if ε → 0, we recover the evolution equation ofMiMe scheme. If on the other hand ε is not too small, then theeffect of realignement is to add an implicit term to the integration ofthe equation for macroscopic moments, thus increasing its stabilityregion.




Conclusion

We have proposed a Microscopically Implicit MacroscopicallyExplicit scheme for the BGK equation. The following are a fewnumerical results I have not shown, otherwise it would take evenlonger...

1 The scheme uses a macroscopic CFL. This corresponds, forthe tests shown, to an increase from 4 to 10 times of the CFLbased on the fastest modes which was characteristic of ourprevious IMEX scheme..

2 The absolute errors are roughly 50% smaller than with thecorresponding IMEX scheme based on the microscopic CFL.

3 The condition number of the coefficient matrix for the solutionof the system for f is small (around 10) and it decreases asthe Knudsen number decreases and CFL is increased.




Conclusion

We have proposed a Microscopically Implicit MacroscopicallyExplicit scheme for the BGK equation. The following are a fewnumerical results I have not shown, otherwise it would take evenlonger...

1 The scheme uses a macroscopic CFL. This corresponds, forthe tests shown, to an increase from 4 to 10 times of the CFLbased on the fastest modes which was characteristic of ourprevious IMEX scheme..

2 The absolute errors are roughly 50% smaller than with thecorresponding IMEX scheme based on the microscopic CFL.

3 The condition number of the coefficient matrix for the solutionof the system for f is small (around 10) and it decreases asthe Knudsen number decreases and CFL is increased.Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models



Perspectives

In the Compressible Navier Stokes regime, we integrate themacroscopic equations with a hyperbolic CFL, which is in generalless restrictive than the parabolic CFL needed by an explicit CNSsolver

1 We think to generalize these results to Compressible Eulerequations with small Mach numbers, decoupling fast modes,which could be solved implicitly using relaxation schemesbased on the BGK approach of Natalini et al., from theremaining part of the equations

2 We think of using this BGK solver in domain decompositionstrategies, where one could use the kinetic solver (here BGK)with the same time step of the Euler (hydrodynamic) solver.This approach has already been partially carried out in Alaia,Puppo, JCP 2012.




Thank you!


implicit schemes for the equation of the bgk model · boltzmann equation for the evolution of a...

Documents