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
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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)
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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:
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.)
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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:
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Outline
The BGK equation and its properties
Numerical difficulties
Microscopically Implicit, Macroscopically Explicit (MiMe)schemes
Numerical examples
Asymptotic properties of MiMe schemes
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Outline
The BGK equation and its properties
Numerical difficulties
Microscopically Implicit, Macroscopically Explicit (MiMe)schemes
Numerical examples
Asymptotic properties of MiMe schemes
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Outline
The BGK equation and its properties
Numerical difficulties
Microscopically Implicit, Macroscopically Explicit (MiMe)schemes
Numerical examples
Asymptotic properties of MiMe schemes
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Outline
The BGK equation and its properties
Numerical difficulties
Microscopically Implicit, Macroscopically Explicit (MiMe)schemes
Numerical examples
Asymptotic properties of MiMe schemes
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Conservation
Since ρρuE
=
⟨f
1v
12 ||v ||
2
⟩ =
⟨fM
1v
12 ||v ||
2
⟩
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical schemes for the BGK model
The development of numerical methods for the BGK model hasstarted only recently.
Aoki, Kanba, Takata ’97This is a second order scheme, designed for smooth solutions
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical schemes for the BGK model
The development of numerical methods for the BGK model hasstarted only recently.
Mieussens, ’00Second order schemes, where conservation is exactly enforced.Both explicit and implicit case are considered.
Bennoune, Lemou, Mieussens, ’08Micro-Macro decomposition
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical schemes for the BGK model
The development of numerical methods for the BGK model hasstarted only recently.
Andries, Bourgat, le Tallec, Perthame ’02A stochastic Monte Carlo scheme
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical schemes for the BGK model
The development of numerical methods for the BGK model hasstarted only recently.
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical schemes for the BGK model
The development of numerical methods for the BGK model hasstarted only recently.
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical macroscopic fluxes
Write the macroscopic moment equations as:
∂tu(f ) = −∂xG(f ),
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical macroscopic fluxes
Write the macroscopic moment equations as:
∂tu(f ) = −∂xG(f ),
where u and G are
u =
〈f 〉
〈fv〉⟨12‖v‖
2f⟩
G =
〈fv〉
〈v ⊗ vf 〉⟨12‖v‖
2vf⟩
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Numerical macroscopic fluxes
Write the macroscopic moment equations as:
∂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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Convergence rate
Convergence rate on macroscopic quantities for Kn = 10−1 andKn = 10−2 in the L1 norm, on a smooth profile.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Realignement of moments-1
MiMe1, Kn = 0.1, temperature profile without (left) and with(right) realignement for several grids
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Realignement of moments-2
MiMe2, Kn = 0.1, temperature profile without (left) and with(right) realignement for several grids
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Changing the order
MiMe, Kn = 10−5, density and temperature profiles with first orderMiMe scheme (cyan), second order MiMe (magenta) and thirdorder explicit (black)
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
).
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Asymptotics for semidiscrete MiMe scheme
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)
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Asymptotics for semidiscrete MiMe scheme
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)
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Asymptotics for semidiscrete MiMe scheme
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)
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Asymptotics for semidiscrete MiMe scheme
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Convergence to CNS
MiMe2. Convergence to Compressible Navier-Stokes. Left to right:Kn = 0.1, Kn = 0.05
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Convergence to CNS
MiMe2. Convergence to Compressible Navier-Stokes. Left to right,Kn = 0.02 and Kn = 0.01
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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...
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
).
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Semidiscrete MiMe with realignement
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
).
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Semidiscrete MiMe with realignement
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
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.
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models
The BGK modelNumerical schemes for BGK models
MiMe numerical schemes for BGK modelsCompressible NS asymptotics
Thank you!
Sandra Pieraccini, Gabriella Puppo Microscopically Implicit Macroscopically Explicit schemes for kinetic models