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Lattice Boltzmann methodsSummer term 2017
Cumulant-based LBMProf. B. Wohlmuth Markus Muhr
25. July 2017
Lattice Boltzmann methodsSummer term 2017
Content1. Repetition: LBM and notation
DiscretizationSRT-equation
2. MRT-modelBasic ideaMoments and moment-transformation
3. Cumulant-based LBMBasic idea and technical toolsDerivation of the cumulantsCumulant-transformationCumulant collision operator
4. Validation a.k.a. colorful pictures
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 1/60
Lattice Boltzmann methodsSummer term 2017
Repetition: LBM and notation
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 2/60
Lattice Boltzmann methodsSummer term 2017
Computational Fluid Dynamics
� Numerical simulation of flows
� Calculation of the velocity field
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 3/60
Lattice Boltzmann methodsSummer term 2017
Computational Fluid Dynamics
� Numerical simulation of flows
� Calculation of the velocity field
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 4/60
Lattice Boltzmann methodsSummer term 2017
Discretization
� In space
� In velocity (D2Q9-model)
c1c3
c2
c4
c5c6
c7 c8
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 5/60
Lattice Boltzmann methodsSummer term 2017
(Discrete) Boltzmann-equation
� Considering „probabilities of an encounter“ fi
f1f3
f2
f4
f5f6
f7 f8
� Derivation of a (discrete) transport-equation for thedistributions fi :
∂fi(t, x)∂t + ci ·∇x fi(t, x) = Q(fi) ≈ −
1τC·(fi(t, x)− f eqi (t, x)
)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 6/60
Lattice Boltzmann methodsSummer term 2017
(Discrete) Boltzmann-equation
� Considering „probabilities of an encounter“ fi
f1f3
f2
f4
f5f6
f7 f8
� Derivation of a (discrete) transport-equation for thedistributions fi :
∂fi(t, x)∂t + ci ·∇x fi(t, x) = Q(fi) ≈ −
1τC·(fi(t, x)− f eqi (t, x)
)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 6/60
Lattice Boltzmann methodsSummer term 2017
Equilibrium distributions f eqi
� Maxwell-distribution as „natural“velocity-distribution of anideal gas:
M(ξ, ρ, u,T ) = ρ ·( m2π kBT
)3/2· exp
(−
m |ξ − u|2
2kBT
)
� Taylor expansion up to second order ξ = ci :
f eqi = ρωi ·(1 +
3 ci · uc2 − 3 |u|2
2c2 + 12
9c4(ci · u
)2)
CKumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 7/60
Lattice Boltzmann methodsSummer term 2017
SRT-equation
� Discretization of the derivative by finite differencesfi(t + ∆t, x + ci∆t) = fi(t, x)− 1
τ·(fi(t, x)− f eqi (t, x)
)� Interpretation as collision and streaming:
f1f3
f2
f4
f5f6
f7 f8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 8/60
Lattice Boltzmann methodsSummer term 2017
SRT-equation
� Discretization of the derivative by finite differencesfi(t + ∆t, x + ci∆t) = fi(t, x)− 1
τ·(fi(t, x)− f eqi (t, x)
)� Interpretation as collision and streaming:
f1f3
f2
f4
f5f6
f7 f8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 8/60
Lattice Boltzmann methodsSummer term 2017
SRT-equation
� Discretization of the derivative by finite differencesfi(t + ∆t, x + ci∆t) = fi(t, x)− 1
τ·(fi(t, x)− f eqi (t, x)
)� Interpretation as collision and streaming:
f1f3
f2
f4
f5f6
f7 f8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 8/60
Lattice Boltzmann methodsSummer term 2017
SRT-equation
� Discretization of the derivative by finite differencesfi(t + ∆t, x + ci∆t) = fi(t, x)− 1
τ·(fi(t, x)− f eqi (t, x)
)� Interpretation as collision and streaming:
f1f3
f2
f4
f5f6
f7 f8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
f ∗1f ∗3
f ∗2
f ∗4
f ∗5f ∗6
f ∗7 f ∗8
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 8/60
Lattice Boltzmann methodsSummer term 2017
In three dimensions
� Analogously in three dimensions (D3Q27-model)
1
2
3
4
5
6
78
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
x
yz
1
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 9/60
Lattice Boltzmann methodsSummer term 2017
Notation� Convenient notation by index-tupel:
f1f3
f2
f4
f5f6
f7 f8
f10f−10
f01
f0−1
f11f−11
f−1−1 f1−1
fi → fij with i , j ∈ {−1, 0, 1}
� Analogously in three dimensions: fi → fijk withi , j , k ∈ {−1, 0, 1}
�Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 10/60
Lattice Boltzmann methodsSummer term 2017
MRT-model
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 11/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of MRT (1)
� The SRT-equation reads:
fi(t + ∆t, x + ci∆t) = fi(t, x)− 1τ·(fi(t, x)− f eqi (t, x)
)� Idea: Mix the relaxation terms
fi(t + ∆t, x + ci∆t) = fi(t, x)−nV∑j=0
1τij·(fj(t, x)− f eqj (t, x)
)
� In matrix-vector-form...
fi(t + ∆t, x + ci∆t)...
=
...
fi(t, x)...
−S·
...(
fi(t, x)− f eqi (t, x))
...
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 12/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of MRT (1)
� The SRT-equation reads:
fi(t + ∆t, x + ci∆t) = fi(t, x)− 1τ·(fi(t, x)− f eqi (t, x)
)� Idea: Mix the relaxation terms
fi(t + ∆t, x + ci∆t) = fi(t, x)−nV∑j=0
1τij·(fj(t, x)− f eqj (t, x)
)� In matrix-vector-form
...fi(t + ∆t, x + ci∆t)
...
=
...
fi(t, x)...
−S·
...(
fi(t, x)− f eqi (t, x))
...
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 12/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of MRT (1)
� The SRT-equation reads:
fi(t + ∆t, x + ci∆t) = fi(t, x)− 1τ·(fi(t, x)− f eqi (t, x)
)� Idea: Mix the relaxation terms
fi(t + ∆t, x + ci∆t) = fi(t, x)−nV∑j=0
1τij·(fj(t, x)− f eqj (t, x)
)� In matrix-vector-form
...fi(t + ∆t, x + ci∆t)
...
=
...
fi(t, x)...
−S·
...(
fi(t, x)− f eqi (t, x))
...
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 12/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of MRT (2)
� In matrix-vector-form
f (t + ∆t, x + c×∆t) = f (t, x)− S · (f (t, x)− f eq(t, x))
� Diagonalize the matrix S:
f (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
� Interpretation as transformation M : f 7→ m
f (t + ∆t, x + c×∆t) = f (t, x)−M−1S · (m(t, x)−meq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 13/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of MRT (2)
� In matrix-vector-form
f (t + ∆t, x + c×∆t) = f (t, x)− S · (f (t, x)− f eq(t, x))
� Diagonalize the matrix S:
f (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
� Interpretation as transformation M : f 7→ m
f (t + ∆t, x + c×∆t) = f (t, x)−M−1S · (m(t, x)−meq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 13/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of MRT (2)
� In matrix-vector-form
f (t + ∆t, x + c×∆t) = f (t, x)− S · (f (t, x)− f eq(t, x))
� Diagonalize the matrix S:
f (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
� Interpretation as transformation M : f 7→ m
f (t + ∆t, x + c×∆t) = f (t, x)−M−1S · (m(t, x)−meq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 13/60
Lattice Boltzmann methodsSummer term 2017
Moments
Definition: (Raw)-momentsFor α, β, γ = 0, 1, 2, ... one defined the (raw)-moments of orderα + β + γ as follows:
mαβγ =1∑
i ,j,k=−1
(cijk)αx·(cijk)βy·(cijk)γz· fijk =
1∑i ,j,k=−1
iα · jβ ·kγ · fijk
� In matrix-vector-form (in 2D)
m00m10m01m20m02m11m21m12m22
=
1 1 1 1 1 1 1 1 10 1 0 −1 0 1 −1 −1 10 0 1 0 −1 1 1 −1 −10 1 0 1 0 1 1 1 10 0 1 0 1 1 1 1 10 0 0 0 0 1 −1 1 −10 0 0 0 0 1 1 −1 −10 0 0 0 0 1 −1 −1 10 0 0 0 0 1 1 1 1
·
f0f1f2f3f4f5f6f7f8
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 14/60
Lattice Boltzmann methodsSummer term 2017
Moments
Definition: (Raw)-momentsFor α, β, γ = 0, 1, 2, ... one defined the (raw)-moments of orderα + β + γ as follows:
mαβγ =1∑
i ,j,k=−1
(cijk)αx·(cijk)βy·(cijk)γz· fijk =
1∑i ,j,k=−1
iα · jβ ·kγ · fijk
� In matrix-vector-form (in 2D)
m00m10m01m20m02m11m21m12m22
=
1 1 1 1 1 1 1 1 10 1 0 −1 0 1 −1 −1 10 0 1 0 −1 1 1 −1 −10 1 0 1 0 1 1 1 10 0 1 0 1 1 1 1 10 0 0 0 0 1 −1 1 −10 0 0 0 0 1 1 −1 −10 0 0 0 0 1 −1 −1 10 0 0 0 0 1 1 1 1
·
f0f1f2f3f4f5f6f7f8
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 14/60
Lattice Boltzmann methodsSummer term 2017
Interpretation of the moments
� The moments correspond to different physical quantities like:
I m00 =1∑
i,j=−1fij = ρ −→ density
c1c3
c2
c4
c5c6
c7 c8
I m10 =1∑
i,j=−1
(cij)x· fij = ρ · ux −→ x-momentum
I m01 =1∑
i,j=−1
(cij)y· fij = ρ · uy −→ y-momentum
� Some moments have to be combined for a reasonableinterpretationI m20 + m02 =
1∑i,j=−1
((cij)2x
+(
cij)2y
)· fij ∼ Ekin
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 15/60
Lattice Boltzmann methodsSummer term 2017
Interpretation of the moments
� The moments correspond to different physical quantities like:
I m00 =1∑
i,j=−1fij = ρ −→ density
c1c3
c2
c4
c5c6
c7 c8
I m10 =1∑
i,j=−1
(cij)x· fij = ρ · ux −→ x-momentum
I m01 =1∑
i,j=−1
(cij)y· fij = ρ · uy −→ y-momentum
� Some moments have to be combined for a reasonableinterpretationI m20 + m02 =
1∑i,j=−1
((cij)2x
+(
cij)2y
)· fij ∼ Ekin
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 15/60
Lattice Boltzmann methodsSummer term 2017
Interpretation of the moments
� The moments correspond to different physical quantities like:
I m00 =1∑
i,j=−1fij = ρ −→ density
c1c3
c2
c4
c5c6
c7 c8
I m10 =1∑
i,j=−1
(cij)x· fij = ρ · ux −→ x-momentum
I m01 =1∑
i,j=−1
(cij)y· fij = ρ · uy −→ y-momentum
� Some moments have to be combined for a reasonableinterpretationI m20 + m02 =
1∑i,j=−1
((cij)2x
+(
cij)2y
)· fij ∼ Ekin
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 15/60
Lattice Boltzmann methodsSummer term 2017
Moment-transformation
� The respective linear combination of matrix-lines yields themoment-transformationmatrix
M0M1M2M3M4M5M6M7M8
=
m00m10m01
m20 + m02m20 − m02
m11m21m12m22
=
1 1 1 1 1 1 1 1 10 1 0 −1 0 1 −1 −1 10 0 1 0 −1 1 1 −1 −10 1 1 1 1 2 2 2 20 1 −1 1 −1 0 0 0 00 0 0 0 0 1 −1 1 −10 0 0 0 0 1 1 −1 −10 0 0 0 0 1 −1 −1 10 0 0 0 0 1 1 1 1
·
f0f1f2f3f4f5f6f7f8
� Further linear combinations of (raw)-moments are possible asfor example row-wise orthogonalization.
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 16/60
Lattice Boltzmann methodsSummer term 2017
MRT-algorithm
� Transform the given distributions f into moments M = M · fund Meq = M · f eq
� Relax the moment Mi with frequencies ωiM∗0...
M∗8
=
ω0
. . .ω8
·
M0 −Meq0
...M8 −Meq
8
� Transform the relaxed moments back into distribution space
f ∗ = M−1 ·M∗
� Collision update and streaming stepf (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 17/60
Lattice Boltzmann methodsSummer term 2017
MRT-algorithm
� Transform the given distributions f into moments M = M · fund Meq = M · f eq
� Relax the moment Mi with frequencies ωiM∗0...
M∗8
=
ω0
. . .ω8
·
M0 −Meq0
...M8 −Meq
8
� Transform the relaxed moments back into distribution spacef ∗ = M−1 ·M∗
� Collision update and streaming stepf (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 17/60
Lattice Boltzmann methodsSummer term 2017
MRT-algorithm
� Transform the given distributions f into moments M = M · fund Meq = M · f eq
� Relax the moment Mi with frequencies ωiM∗0...
M∗8
=
ω0
. . .ω8
·
M0 −Meq0
...M8 −Meq
8
� Transform the relaxed moments back into distribution space
f ∗ = M−1 ·M∗
� Collision update and streaming stepf (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 17/60
Lattice Boltzmann methodsSummer term 2017
MRT-algorithm
� Transform the given distributions f into moments M = M · fund Meq = M · f eq
� Relax the moment Mi with frequencies ωiM∗0...
M∗8
=
ω0
. . .ω8
·
M0 −Meq0
...M8 −Meq
8
� Transform the relaxed moments back into distribution space
f ∗ = M−1 ·M∗
� Collision update and streaming stepf (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 17/60
Lattice Boltzmann methodsSummer term 2017
MRT-algorithm
� Transform the given distributions f into moments M = M · fund Meq = M · f eq
� Relax the moment Mi with frequencies ωiM∗0...
M∗8
=
ω0
. . .ω8
·
M0 −Meq0
...M8 −Meq
8
� Transform the relaxed moments back into distribution space
f ∗ = M−1 ·M∗
� Collision update and streaming stepf (t+∆t, x +c×∆t) = f (t, x)−M−1S ·(Mf (t, x)−Mf eq(t, x))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 17/60
Lattice Boltzmann methodsSummer term 2017
Pro/Con MRT-model� More „tuning-parameter“ ωi as for SRT� Better insight into the model by identification with physicalquantities
� Linear transformations
� statistical dependencies among the moments(→ dependencies between ωi)
� Physics gets lost by (wrong) variation of parameters. How tochoose them?
�Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 18/60
Lattice Boltzmann methodsSummer term 2017
Pro/Con MRT-model� More „tuning-parameter“ ωi as for SRT� Better insight into the model by identification with physicalquantities
� Linear transformations
� statistical dependencies among the moments(→ dependencies between ωi)
� Physics gets lost by (wrong) variation of parameters. How tochoose them?
�Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 18/60
Lattice Boltzmann methodsSummer term 2017
Cumulant-based LBM
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 19/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of cumulant-based LBM
� Goal: Find a transformation to statistically independentquantities Cm
� Mathematically this is described by a factorization of theprobability-density
fijk =nV∏m=0
Fm(Cm)
� Efficient implementation of that (nonlinear) transformation
� Main tools: Multivariate Taylor-expansion,Laplace-transformation and distribution-theory
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 20/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of cumulant-based LBM
� Goal: Find a transformation to statistically independentquantities Cm
� Mathematically this is described by a factorization of theprobability-density
fijk =nV∏m=0
Fm(Cm)
� Efficient implementation of that (nonlinear) transformation� Main tools: Multivariate Taylor-expansion,
Laplace-transformation and distribution-theory
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 20/60
Lattice Boltzmann methodsSummer term 2017
Basic idea of cumulant-based LBM
� Goal: Find a transformation to statistically independentquantities Cm
� Mathematically this is described by a factorization of theprobability-density
fijk =nV∏m=0
Fm(Cm)
� Efficient implementation of that (nonlinear) transformation� Main tools: Multivariate Taylor-expansion,
Laplace-transformation and distribution-theory
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 20/60
Lattice Boltzmann methodsSummer term 2017
Distributions� Recall from measure theory: Lebesgue-space L2:
L2(R) :={f : R→ R |
∫R |f (x)|2 dx <∞
}� Consider the following functional
G : L2(R)→ R, G(f ) :=∫ 1
−1f (x) dx
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 21/60
Lattice Boltzmann methodsSummer term 2017
Distributions� Recall from measure theory: Lebesgue-space L2:
L2(R) :={f : R→ R |
∫R |f (x)|2 dx <∞
}� Consider the following functional
G : L2(R)→ R, G(f ) :=∫ 1
−1f (x) dx
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 21/60
Lattice Boltzmann methodsSummer term 2017
Distributions� Recall from measure theory: Lebesgue-space L2:
L2(R) :={f : R→ R |
∫R |f (x)|2 dx <∞
}� Consider the following functional
G : L2(R)→ R, G(f ) :=∫ 1
−1f (x) dx
� One searches for a g ∈ L2(R) satisfying:
G(f ) =∫R
g(x) · f (x) dx ∀f ∈ L2(R)
� Solution: g(x) = 1[−1,1](x)
� Interpretation of g as functional (Riesz representationtheorem)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 22/60
Lattice Boltzmann methodsSummer term 2017
Distributions� Recall from measure theory: Lebesgue-space L2:
L2(R) :={f : R→ R |
∫R |f (x)|2 dx <∞
}� Consider the following functional
G : L2(R)→ R, G(f ) :=∫ 1
−1f (x) dx
� One searches for a g ∈ L2(R) satisfying:
G(f ) =∫R
g(x) · f (x) dx ∀f ∈ L2(R)
� Solution: g(x) = 1[−1,1](x)
� Interpretation of g as functional (Riesz representationtheorem)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 22/60
Lattice Boltzmann methodsSummer term 2017
Distributions� Recall from measure theory: Lebesgue-space L2:
L2(R) :={f : R→ R |
∫R |f (x)|2 dx <∞
}� Consider the following functional
G : L2(R)→ R, G(f ) :=∫ 1
−1f (x) dx
� One searches for a g ∈ L2(R) satisfying:
G(f ) =∫R
g(x) · f (x) dx ∀f ∈ L2(R)
� Solution: g(x) = 1[−1,1](x)
� Interpretation of g as functional (Riesz representationtheorem)Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 22/60
Lattice Boltzmann methodsSummer term 2017
Distributions� Recall from measure theory: Lebesgue-space L2:
L2(R) :={f : R→ R |
∫R |f (x)|2 dx <∞
}� Consider the following functional
G : L2(R)→ R, G(f ) :=∫ 1
−1f (x) dx
� One searches for a g ∈ L2(R) satisfying:
G(f ) =∫R
g(x) · f (x) dx ∀f ∈ L2(R)
� Solution: g(x) = 1[−1,1](x)
� Interpretation of g as functional (Riesz representationtheorem)Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 22/60
Lattice Boltzmann methodsSummer term 2017
Dirac-sequence
� Interprete the functions gn(x) = n2 · 1[− 1
n ,1n ](x), n ∈ N as
functionals
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 23/60
Lattice Boltzmann methodsSummer term 2017
Delta-distribution δ
� For continuous functions f there holds:∫R
gn(x) · f (x) dx = n2 ·∫ 1
n
− 1n
f (x) dx
= n2 ·(
F(1
n
)− F
(−1
n
))
=F(1n
)− F
(− 1
n
)2 · 1n
n→∞−→ F ′(0) = f (0)
� Define the delta-distribution δ(x) := limn→∞
gn(x) = „g∞(x)“by: ∫
Rδ(x) · f (x) dx = f (0)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 24/60
Lattice Boltzmann methodsSummer term 2017
Delta-distribution δ
� For continuous functions f there holds:∫R
gn(x) · f (x) dx = n2 ·∫ 1
n
− 1n
f (x) dx
= n2 ·(
F(1
n
)− F
(−1
n
))
=F(1n
)− F
(− 1
n
)2 · 1n
n→∞−→ F ′(0) = f (0)
� Define the delta-distribution δ(x) := limn→∞
gn(x) = „g∞(x)“by: ∫
Rδ(x) · f (x) dx = f (0)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 24/60
Lattice Boltzmann methodsSummer term 2017
Delta-distribution δ
� For continuous functions f there holds:∫R
gn(x) · f (x) dx = n2 ·∫ 1
n
− 1n
f (x) dx
= n2 ·(
F(1
n
)− F
(−1
n
))
=F(1n
)− F
(− 1
n
)2 · 1n
n→∞−→ F ′(0) = f (0)
� Define the delta-distribution δ(x) := limn→∞
gn(x) = „g∞(x)“by: ∫
Rδ(x) · f (x) dx = f (0)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 24/60
Lattice Boltzmann methodsSummer term 2017
Delta-distribution δ
� For continuous functions f there holds:∫R
gn(x) · f (x) dx = n2 ·∫ 1
n
− 1n
f (x) dx
= n2 ·(
F(1
n
)− F
(−1
n
))
=F(1n
)− F
(− 1
n
)2 · 1n
n→∞−→ F ′(0) = f (0)
� Define the delta-distribution δ(x) := limn→∞
gn(x) = „g∞(x)“by: ∫
Rδ(x) · f (x) dx = f (0)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 24/60
Lattice Boltzmann methodsSummer term 2017
Delta-distribution δ
� For continuous functions f there holds:∫R
gn(x) · f (x) dx = n2 ·∫ 1
n
− 1n
f (x) dx
= n2 ·(
F(1
n
)− F
(−1
n
))
=F(1n
)− F
(− 1
n
)2 · 1n
n→∞−→ F ′(0) = f (0)
� Define the delta-distribution δ(x) := limn→∞
gn(x) = „g∞(x)“by: ∫
Rδ(x) · f (x) dx = f (0)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 24/60
Lattice Boltzmann methodsSummer term 2017
Delta-function
� Considered as a function there holds: δ(x) ={0 if x 6= 0∞ if x = 0
� Which often gets normed to: δ(x) ={0 if x 6= 01 if x = 0
� What you should remember for sure:∫Rδ(x) · f (x) dx = f (0)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 25/60
Lattice Boltzmann methodsSummer term 2017
Delta-function
� Considered as a function there holds: δ(x) ={0 if x 6= 0∞ if x = 0
� Which often gets normed to: δ(x) ={0 if x 6= 01 if x = 0
� What you should remember for sure:∫Rδ(x) · f (x) dx = f (0)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 25/60
Lattice Boltzmann methodsSummer term 2017
Repetition: Fourier-transform� The Fourier-transform of f : R→ C is (modulo scaling)defined by:
f (ω) = F [f (x)](ω) :=∫R
f (x) · e−iωx dx
� f (ω) tells, how strong eiω is present in thefrequency-spectrum of f
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 26/60
Lattice Boltzmann methodsSummer term 2017
Repetition: Fourier-transform� The Fourier-transform of f : R→ C is (modulo scaling)defined by:
f (ω) = F [f (x)](ω) :=∫R
f (x) · e−iωx dx
� f (ω) tells, how strong eiω is present in thefrequency-spectrum of f
� Example: F [sin(x)](ω) ∼ δ(ω − 1)− δ(ω + 1)2i
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 27/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation
� Instead of projection onto oscillations {eiω}ω∈R one„projects“ onto damping-functions {e−s}s∈R+
F (s) := L[f (x)](s) :=∫ ∞0
f (x) · e−sx dx
� Laplace-transformations of a few common functions:I L[xn](s) = n!
sn+1
I L[sin(ax)](s) = as2+a2
I L[ln(ax)](s) = − 1s (ln(s/a) + γ)
� Certain features as for example the shift-property hold true:L[f (x − a)](s) = e−asF (s)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 28/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation
� Instead of projection onto oscillations {eiω}ω∈R one„projects“ onto damping-functions {e−s}s∈R+
F (s) := L[f (x)](s) :=∫ ∞0
f (x) · e−sx dx
� Laplace-transformations of a few common functions:I L[xn](s) = n!
sn+1
I L[sin(ax)](s) = as2+a2
I L[ln(ax)](s) = − 1s (ln(s/a) + γ)
� Certain features as for example the shift-property hold true:L[f (x − a)](s) = e−asF (s)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 28/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation
� Instead of projection onto oscillations {eiω}ω∈R one„projects“ onto damping-functions {e−s}s∈R+
F (s) := L[f (x)](s) :=∫ ∞0
f (x) · e−sx dx
� Laplace-transformations of a few common functions:I L[xn](s) = n!
sn+1
I L[sin(ax)](s) = as2+a2
I L[ln(ax)](s) = − 1s (ln(s/a) + γ)
� Certain features as for example the shift-property hold true:L[f (x − a)](s) = e−asF (s)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 28/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation of the delta-distribution
� Apply the definition of the (two-sided) Laplace-transform:
L[δ(x)](s) =∫Rδ(x) · e−sx dx
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 29/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation of the delta-distribution
� Apply the definition of the (two-sided) Laplace-transform:
L[δ(x)](s) =∫Rδ(x) · e−sx dx
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 29/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation of the delta-distribution
� Apply the definition of the (two-sided) Laplace-transform:
L[δ(x)](s) =∫Rδ(x) · e−sx dx = e−s·0 = 1
� Transformation of the shifted delta-distribution by theshift-property:
L[δ(x − a)](s) = e−asL[δ(x)](s)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 30/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation of the delta-distribution
� Apply the definition of the (two-sided) Laplace-transform:
L[δ(x)](s) =∫Rδ(x) · e−sx dx = e−s·0 = 1
� Transformation of the shifted delta-distribution by theshift-property:
L[δ(x − a)](s) = e−asL[δ(x)](s)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 30/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation of the delta-distribution
� Apply the definition of the (two-sided) Laplace-transform:
L[δ(x)](s) =∫Rδ(x) · e−sx dx = e−s·0 = 1
� Transformation of the shifted delta-distribution by theshift-property:
L[δ(x − a)](s) = e−asL[δ(x)](s)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 30/60
Lattice Boltzmann methodsSummer term 2017
Laplace-transformation of the delta-distribution
� Apply the definition of the (two-sided) Laplace-transform:
L[δ(x)](s) =∫Rδ(x) · e−sx dx = e−s·0 = 1
� Transformation of the shifted delta-distribution by theshift-property:
L[δ(x − a)](s) = e−asL[δ(x)](s) = e−as
♦Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 31/60
Lattice Boltzmann methodsSummer term 2017
Derivation of the cumulants (1) - „Analytization“
� Want to capture the discrete velocity-distributions analytically
f10f−10
f01
f0−1
f11f−11
f−1−1 f1−1
� This can be done by the „peak-function“ f (ξ):
f (ξ) = f (ξx , ξy , ξz) =1∑
i ,j,k=−1fijk δ(ic−ξx ) δ(jc−ξy ) δ(kc−ξz)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 32/60
Lattice Boltzmann methodsSummer term 2017
Derivation of the cumulants (1) - „Analytization“
� Want to capture the discrete velocity-distributions analytically
f10f−10
f01
f0−1
f11f−11
f−1−1 f1−1
� This can be done by the „peak-function“ f (ξ):
f (ξ) = f (ξx , ξy , ξz) =1∑
i ,j,k=−1fijk δ(ic−ξx ) δ(jc−ξy ) δ(kc−ξz)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 32/60
Lattice Boltzmann methodsSummer term 2017
Derivation of the cumulants (1) - „Analytization“
� Want to capture the discrete velocity-distributions analytically
f10f−10
f01
f0−1
f11f−11
f−1−1 f1−1
� This can be done by the „peak-function“ f (ξ):
f (ξ) = f (ξx , ξy , ξz) =1∑
i ,j,k=−1fijk δ(ic−ξx ) δ(jc−ξy ) δ(kc−ξz)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 33/60
Lattice Boltzmann methodsSummer term 2017
Derivation (2) - Laplace-transform
� To increase regularity one performs the two-sidedLaplace-transform of f (ξ):
L[f (ξ)](Ξ) =1∑
i ,j,k=−1fijkL[δ(ic−ξx )]L[δ(jc−ξy )]L[δ(kc−ξz)]
=1∑
i ,j,k=−1fijk e−Ξx ic e−Ξy jc e−Ξzkc =: F (Ξ)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 34/60
Lattice Boltzmann methodsSummer term 2017
Derivation (2) - Laplace-transform
� To increase regularity one performs the two-sidedLaplace-transform of f (ξ):
L[f (ξ)](Ξ) =1∑
i ,j,k=−1fijkL[δ(ic−ξx )]L[δ(jc−ξy )]L[δ(kc−ξz)]
=1∑
i ,j,k=−1fijk e−Ξx ic e−Ξy jc e−Ξzkc =: F (Ξ)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 34/60
Lattice Boltzmann methodsSummer term 2017
Derivation (3) - statistical independence
� Applying the statistical independence (which one aims tohave) in the frequency space:
F (Ξx ,Ξy ,Ξz) =nV∏m=0
Fm(Cm)
� And using the logarithm
ln (F (Ξx ,Ξy ,Ξz)) =nV∑m=0
ln (Fm(Cm))
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 35/60
Lattice Boltzmann methodsSummer term 2017
Derivation (4) - Taylor-expansion
� The multivariate Taylor-expansion of ln (F (Ξx ,Ξy ,Ξz))around Ξ = 0 is:
ln(F (Ξx ,Ξy ,Ξz )) =∑
α+β+γ≥0
1α!β!γ!
∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγzln(F (Ξx ,Ξy ,Ξz ))
∣∣∣∣Ξ=0
· Ξαx Ξβy Ξγz
Definition: CumulantsFor α, β, γ = 0, 1, 2, ... one defines the cumulants of orderα + β + γ as follows:
cαβγ := c−(α+β+γ) ∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγzln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 36/60
Lattice Boltzmann methodsSummer term 2017
Derivation (4) - Taylor-expansion
� The multivariate Taylor-expansion of ln (F (Ξx ,Ξy ,Ξz))around Ξ = 0 is:
ln(F (Ξx ,Ξy ,Ξz )) =∑
α+β+γ≥0
1α!β!γ!
∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγzln(F (Ξx ,Ξy ,Ξz ))
∣∣∣∣Ξ=0
· Ξαx Ξβy Ξγz
Definition: CumulantsFor α, β, γ = 0, 1, 2, ... one defines the cumulants of orderα + β + γ as follows:
cαβγ := c−(α+β+γ) ∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγzln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 36/60
Lattice Boltzmann methodsSummer term 2017
Efficient calculation via moments
� Conduct the Taylor-expansion without the logarithm:
F (Ξx ,Ξy ,Ξz ) =∑
α+β+γ≥0
1α!β!γ!
∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz )
∣∣∣∣Ξ=0
· Ξαx Ξβy Ξγz
� Analogously consider the coefficients:
∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz)
∣∣∣∣∣Ξ=0
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 37/60
Lattice Boltzmann methodsSummer term 2017
Where the magic happens
� Explicit calculation of these coefficients yields:
∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz )
∣∣∣∣Ξ=0
=∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγz
1∑i,j,k=−1
fijk e−Ξx ic e−Ξy jc e−Ξzkc
∣∣∣∣∣Ξ=0
=1∑
i,j,k=−1
fijk (−ic)αe−Ξx ic (−jc)βe−Ξy jc · (−kc)γe−Ξzkc∣∣
Ξ=0
= (−c)α+β+γ1∑
i,j,k=−1
iα jβ kγ fijk
= (−c)α+β+γ mαβγ
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 38/60
Lattice Boltzmann methodsSummer term 2017
Where the magic happens
� Explicit calculation of these coefficients yields:
∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz )
∣∣∣∣Ξ=0
=∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγz
1∑i,j,k=−1
fijk e−Ξx ic e−Ξy jc e−Ξzkc
∣∣∣∣∣Ξ=0
=1∑
i,j,k=−1
fijk (−ic)αe−Ξx ic (−jc)βe−Ξy jc · (−kc)γe−Ξzkc∣∣
Ξ=0
= (−c)α+β+γ1∑
i,j,k=−1
iα jβ kγ fijk
= (−c)α+β+γ mαβγ
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 38/60
Lattice Boltzmann methodsSummer term 2017
Where the magic happens
� Explicit calculation of these coefficients yields:
∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz )
∣∣∣∣Ξ=0
=∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγz
1∑i,j,k=−1
fijk e−Ξx ic e−Ξy jc e−Ξzkc
∣∣∣∣∣Ξ=0
=1∑
i,j,k=−1
fijk (−ic)αe−Ξx ic (−jc)βe−Ξy jc · (−kc)γe−Ξzkc∣∣
Ξ=0
= (−c)α+β+γ1∑
i,j,k=−1
iα jβ kγ fijk
= (−c)α+β+γ mαβγ
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 38/60
Lattice Boltzmann methodsSummer term 2017
Where the magic happens
� Explicit calculation of these coefficients yields:
∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz )
∣∣∣∣Ξ=0
=∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγz
1∑i,j,k=−1
fijk e−Ξx ic e−Ξy jc e−Ξzkc
∣∣∣∣∣Ξ=0
=1∑
i,j,k=−1
fijk (−ic)αe−Ξx ic (−jc)βe−Ξy jc · (−kc)γe−Ξzkc∣∣
Ξ=0
= (−c)α+β+γ1∑
i,j,k=−1
iα jβ kγ fijk
= (−c)α+β+γ mαβγ
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 38/60
Lattice Boltzmann methodsSummer term 2017
Cumulants vs. Moments
� The (raw)-moments were shown to be:
mαβγ = (−c)−(α+β+γ) ∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz)
∣∣∣∣∣Ξ=0
� The cumulants:
cαβγ = c−(α+β+γ) ∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγzln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
� Idea: By applying the chain rule one should be able to traceback the cumulants to the moments...
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 39/60
Lattice Boltzmann methodsSummer term 2017
Cumulants vs. Moments
� The (raw)-moments were shown to be:
mαβγ = (−c)−(α+β+γ) ∂α+β+γ
∂Ξαx ∂Ξβy ∂ΞγzF (Ξx ,Ξy ,Ξz)
∣∣∣∣∣Ξ=0
� The cumulants:
cαβγ = c−(α+β+γ) ∂α+β+γ
∂Ξαx ∂Ξβy ∂Ξγzln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
� Idea: By applying the chain rule one should be able to traceback the cumulants to the moments...
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 39/60
Lattice Boltzmann methodsSummer term 2017
Cumulant-transformation (1)
� We conduct the idea on the example c100:
c100 = c−(1+0+0) ∂1
∂Ξ1x ∂Ξ0
y ∂Ξ0zln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
∂1
∂Ξ1x
F (Ξx ,Ξy ,Ξz)∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
m100(−c)−1 = − 1
F (0, 0, 0)m100
� With F (0, 0, 0) =1∑
i ,j,k=−1fijke0e0e0 = ρ also: c100 = −1
ρm100
� Define: Cαβγ = ρcαβγ , then =⇒ C100 = −m100
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 40/60
Lattice Boltzmann methodsSummer term 2017
Cumulant-transformation (1)
� We conduct the idea on the example c100:
c100 = c−(1+0+0) ∂1
∂Ξ1x ∂Ξ0
y ∂Ξ0zln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
∂1
∂Ξ1x
F (Ξx ,Ξy ,Ξz)∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
m100(−c)−1 = − 1
F (0, 0, 0)m100
� With F (0, 0, 0) =1∑
i ,j,k=−1fijke0e0e0 = ρ also: c100 = −1
ρm100
� Define: Cαβγ = ρcαβγ , then =⇒ C100 = −m100
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 40/60
Lattice Boltzmann methodsSummer term 2017
Cumulant-transformation (1)
� We conduct the idea on the example c100:
c100 = c−(1+0+0) ∂1
∂Ξ1x ∂Ξ0
y ∂Ξ0zln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
∂1
∂Ξ1x
F (Ξx ,Ξy ,Ξz)∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
m100(−c)−1 = − 1
F (0, 0, 0)m100
� With F (0, 0, 0) =1∑
i ,j,k=−1fijke0e0e0 = ρ also: c100 = −1
ρm100
� Define: Cαβγ = ρcαβγ , then =⇒ C100 = −m100
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 40/60
Lattice Boltzmann methodsSummer term 2017
Cumulant-transformation (1)
� We conduct the idea on the example c100:
c100 = c−(1+0+0) ∂1
∂Ξ1x ∂Ξ0
y ∂Ξ0zln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
∂1
∂Ξ1x
F (Ξx ,Ξy ,Ξz)∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
m100(−c)−1 = − 1
F (0, 0, 0)m100
� With F (0, 0, 0) =1∑
i ,j,k=−1fijke0e0e0 = ρ also: c100 = −1
ρm100
� Define: Cαβγ = ρcαβγ , then =⇒ C100 = −m100
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 40/60
Lattice Boltzmann methodsSummer term 2017
Cumulant-transformation (1)
� We conduct the idea on the example c100:
c100 = c−(1+0+0) ∂1
∂Ξ1x ∂Ξ0
y ∂Ξ0zln(F (Ξx ,Ξy ,Ξz))
∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
∂1
∂Ξ1x
F (Ξx ,Ξy ,Ξz)∣∣∣∣∣Ξ=0
= c−1 1F (0, 0, 0) ·
m100(−c)−1 = − 1
F (0, 0, 0)m100
� With F (0, 0, 0) =1∑
i ,j,k=−1fijke0e0e0 = ρ also: c100 = −1
ρm100
� Define: Cαβγ = ρcαβγ , then =⇒ C100 = −m100
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 40/60
Lattice Boltzmann methodsSummer term 2017
Cumulant-transformation (2)
� Analogous procedure for the remaining cumulants:
C100 = −m100 C010 = −m010 C001 = −m001
C200 = m200−1ρ
m2100 C020 = m020−
1ρ
m2010 C002 = m002−
1ρ
m2001
C110 = m110−1ρ
m100m010 C101 = m101−1ρ
m100m001 . . .
C210 = −m210 −2ρ2
m2100m010 + 2
ρm110m100 + 1
ρm200m010
...
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 41/60
Lattice Boltzmann methodsSummer term 2017
Half-time score
� What do we have so far? How far do we come with that,starting with the current distributions?I Transform distributions f to (raw)-moments m = M · fI Transform (raw)-moments m to cumulants
C = K(m) = K(M · f ) = C(f )
f (t+∆t, x +c×∆t) = f (t, x)−C−1[S · (C(f (t, x))− C(f eq(t, x)))
]� Efficient transformation of equilibria� Choice of relaxation parameters ωi� Inverse mapping C−1
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 42/60
Lattice Boltzmann methodsSummer term 2017
Cumulant equilibria - Derivation
� The equilibria can be calculated in advance
� Laplace-transformation of Maxwell-distribution:
F eq(Ξ) = L[M(ξ)
](Ξ) =
∫∫∫R
M(ξ) · e−Ξ·ξ dξx dξy dξz
= ρ√C· exp
(14C (Ξ2
x + Ξ2y + Ξ2
z)− Ξx ux − Ξy uy − Ξz uz)
� Apply the logarithm:
ln(F eq(Ξ)) = ln(ρ
ρ0
)−Ξx ux −Ξy uy −Ξz uz + c2s
2(Ξ2x + Ξ2
y + Ξ2z)
� Taylor-expansion can be skipped, since this is a polynomialanyway
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 43/60
Lattice Boltzmann methodsSummer term 2017
Cumulant equilibria - Derivation
� The equilibria can be calculated in advance� Laplace-transformation of Maxwell-distribution:
F eq(Ξ) = L[M(ξ)
](Ξ) =
∫∫∫R
M(ξ) · e−Ξ·ξ dξx dξy dξz
= ρ√C· exp
(14C (Ξ2
x + Ξ2y + Ξ2
z)− Ξx ux − Ξy uy − Ξz uz)
� Apply the logarithm:
ln(F eq(Ξ)) = ln(ρ
ρ0
)−Ξx ux −Ξy uy −Ξz uz + c2s
2(Ξ2x + Ξ2
y + Ξ2z)
� Taylor-expansion can be skipped, since this is a polynomialanyway
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 43/60
Lattice Boltzmann methodsSummer term 2017
Cumulant equilibria - Derivation
� The equilibria can be calculated in advance� Laplace-transformation of Maxwell-distribution:
F eq(Ξ) = L[M(ξ)
](Ξ) =
∫∫∫R
M(ξ) · e−Ξ·ξ dξx dξy dξz
= ρ√C· exp
(14C (Ξ2
x + Ξ2y + Ξ2
z)− Ξx ux − Ξy uy − Ξz uz)
� Apply the logarithm:
ln(F eq(Ξ)) = ln(ρ
ρ0
)−Ξx ux −Ξy uy −Ξz uz + c2s
2(Ξ2x + Ξ2
y + Ξ2z)
� Taylor-expansion can be skipped, since this is a polynomialanyway
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 43/60
Lattice Boltzmann methodsSummer term 2017
Cumulant equilibria - Derivation
� The equilibria can be calculated in advance� Laplace-transformation of Maxwell-distribution:
F eq(Ξ) = L[M(ξ)
](Ξ) =
∫∫∫R
M(ξ) · e−Ξ·ξ dξx dξy dξz
= ρ√C· exp
(14C (Ξ2
x + Ξ2y + Ξ2
z)− Ξx ux − Ξy uy − Ξz uz)
� Apply the logarithm:
ln(F eq(Ξ)) = ln(ρ
ρ0
)−Ξx ux −Ξy uy −Ξz uz + c2s
2(Ξ2x + Ξ2
y + Ξ2z)
� Taylor-expansion can be skipped, since this is a polynomialanywayKumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 43/60
Lattice Boltzmann methodsSummer term 2017
Cumulant equilibria - Derivation
� The equilibria can be calculated in advance� Laplace-transformation of Maxwell-distribution:
F eq(Ξ) = L[M(ξ)
](Ξ) =
∫∫∫R
M(ξ) · e−Ξ·ξ dξx dξy dξz
= ρ√C· exp
(14C (Ξ2
x + Ξ2y + Ξ2
z)− Ξx ux − Ξy uy − Ξz uz)
� Apply the logarithm:
ln(F eq(Ξ)) = ln(ρ
ρ0
)−Ξx ux −Ξy uy −Ξz uz + c2s
2(Ξ2x + Ξ2
y + Ξ2z)
� Taylor-expansion can be skipped, since this is a polynomialanywayKumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 43/60
Lattice Boltzmann methodsSummer term 2017
Cumulant equilibria
� Only a certain few cumulants have an equilibrium not equal 0:
ceq000 = ln(ρ/ρ0)ceq100 = −c−1uxceq010 = −c−1uyceq001 = −c−1uzceq200 = c−2c2sceq020 = c−2c2sceq002 = c−2c2s
� For all others, there holds: ceqαβγ = 0
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 44/60
Lattice Boltzmann methodsSummer term 2017
Half-time score
� What do we have so far? How far do we come with that,starting with the current distributions?I Transform distributions f to (raw)-moments m = M · fI Transform (raw)-moments m to cumulants
C = K(m) = K(M · f ) = C(f )
f (t+∆t, x +c×∆t) = f (t, x)−C−1[S · (C(f (t, x))− C(f eq(t, x)))
]
� Efficient transformation of equilibria� Choice of relaxation parameters ωi� Inverse mapping C−1
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 45/60
Lattice Boltzmann methodsSummer term 2017
Choice of relaxation parameters
� For the most cumulants the equilibrium is 0, hence one getsfor example:
C∗222 = ω10 (C222 − 0)
C∗110 = ω1 C110
C∗101 = ω1 C101
C∗011 = ω1 C011
C∗211 = ω8 C211
C∗121 = ω8 C121
C∗112 = ω8 C112
� Those with non vanishing equilibrium one can combine:
C∗200 − C∗020 = ω1 (C200 − C020)C∗200 − C∗002 = ω1 (C200 − C002)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 46/60
Lattice Boltzmann methodsSummer term 2017
Choice of relaxation parameters
� For the most cumulants the equilibrium is 0, hence one getsfor example:
C∗222 = ω10 (C222 − 0)
C∗110 = ω1 C110
C∗101 = ω1 C101
C∗011 = ω1 C011
C∗211 = ω8 C211
C∗121 = ω8 C121
C∗112 = ω8 C112
� Those with non vanishing equilibrium one can combine:
C∗200 − C∗020 = ω1 (C200 − C020)C∗200 − C∗002 = ω1 (C200 − C002)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 46/60
Lattice Boltzmann methodsSummer term 2017
Choice of relaxation parameters
� For the most cumulants the equilibrium is 0, hence one getsfor example:
C∗222 = ω10 (C222 − 0)
C∗110 = ω1 C110
C∗101 = ω1 C101
C∗011 = ω1 C011
C∗211 = ω8 C211
C∗121 = ω8 C121
C∗112 = ω8 C112
� Those with non vanishing equilibrium one can combine:
C∗200 − C∗020 = ω1 (C200 − C020)C∗200 − C∗002 = ω1 (C200 − C002)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 46/60
Lattice Boltzmann methodsSummer term 2017
Choice of relaxation parameters
� For the most cumulants the equilibrium is 0, hence one getsfor example:
C∗222 = ω10 (C222 − 0)
C∗110 = ω1 C110
C∗101 = ω1 C101
C∗011 = ω1 C011
C∗211 = ω8 C211
C∗121 = ω8 C121
C∗112 = ω8 C112
� Those with non vanishing equilibrium one can combine:
C∗200 − C∗020 = ω1 (C200 − C020)C∗200 − C∗002 = ω1 (C200 − C002)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 46/60
Lattice Boltzmann methodsSummer term 2017
Half-time score
� What do we have so far? How far do we come with that,starting with the current distributions?I Transform distributions f to (raw)-moments m = M · fI Transform (raw)-moments m to cumulants
C = K(m) = K(M · f ) = C(f )
f (t+∆t, x +c×∆t) = f (t, x)−C−1[S · (C(f (t, x))− C(f eq(t, x)))
]
� Efficient transformation of equilibria
� Choice of relaxation parameters ωi� Inverse mapping C−1
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 47/60
Lattice Boltzmann methodsSummer term 2017
Half-time score
� What do we have so far? How far do we come with that,starting with the current distributions?I Transform distributions f to (raw)-moments m = M · fI Transform (raw)-moments m to cumulants
C = K(m) = K(M · f ) = C(f )
f (t+∆t, x +c×∆t) = f (t, x)−C−1[S · (C(f (t, x))− C(f eq(t, x)))
]
� Efficient transformation of equilibria
� Choice of relaxation parameters ωi� Inverse mapping C−1
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 48/60
Lattice Boltzmann methodsSummer term 2017
Advantages of Cumulant-based LBM
� Better representation of physical processes due to statisticalindependence
� Equilibria of cumulants are almost all 0� More (real) degrees of freedom than in classical MRT� Gradual improvement of the method by asymptotic analysisand elimination of error-terms
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 49/60
Lattice Boltzmann methodsSummer term 2017
Synthesis of Navier-Stokes-equation
� By a Chapman-Enskog expansion one can derive theNavier-Stokes equations:
∇ · u = 0∂u∂t + (u · ∇) u = −1
ρ∇p + 1
3
( 1ω1− 1
2
)︸ ︷︷ ︸
=ν
∆u + fρ
� With ω1 one can simulate fluids of different viscosity� Only ω1 directly influences the solution
� Further parameters influence higher order error terms (setthem to 1 in the following)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 50/60
Lattice Boltzmann methodsSummer term 2017
Synthesis of Navier-Stokes-equation
� By a Chapman-Enskog expansion one can derive theNavier-Stokes equations:
∇ · u = 0∂u∂t + (u · ∇) u = −1
ρ∇p + 1
3
( 1ω1− 1
2
)︸ ︷︷ ︸
=ν
∆u + fρ
� With ω1 one can simulate fluids of different viscosity� Only ω1 directly influences the solution� Further parameters influence higher order error terms (setthem to 1 in the following)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 50/60
Lattice Boltzmann methodsSummer term 2017
Synthesis of Navier-Stokes-equation
� By a Chapman-Enskog expansion one can derive theNavier-Stokes equations:
∇ · u = 0∂u∂t + (u · ∇) u = −1
ρ∇p + 1
3
( 1ω1− 1
2
)︸ ︷︷ ︸
=ν
∆u + fρ
� With ω1 one can simulate fluids of different viscosity� Only ω1 directly influences the solution� Further parameters influence higher order error terms (setthem to 1 in the following)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 50/60
Lattice Boltzmann methodsSummer term 2017
Validation a.k.a. colorfulpictures
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 51/60
Lattice Boltzmann methodsSummer term 2017
Turbulence measure� The Reynolds-number is dimensionless parameter of a flow
Re = U · Lν
� At some specific threshold Re∗ the laminar-turbulenttransition starts
� Traditional LBM methods have problems with highReynolds-numbers:
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 52/60
Lattice Boltzmann methodsSummer term 2017
Comparison: BGK ↔ Kumulanten
� Comparison SRT- and cumulant-based LBM for a cylinderflow at Re = 8000
a) BGK-SRT-LBM (Picture source [5]) b) Cumulant-based LBM (Picture source [5])
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 53/60
Lattice Boltzmann methodsSummer term 2017
Turbulence resolution in 3D
� Representation of (turbulence) vortices around a sphere flow:
(Picture source [5])
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 54/60
Lattice Boltzmann methodsSummer term 2017
Turbulence resolution in 3D
(Picture source [5])
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 55/60
Lattice Boltzmann methodsSummer term 2017
Turbulence resolution in 3D
(Picture source [5])
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 56/60
Lattice Boltzmann methodsSummer term 2017
The highlights
� MRT-like-method with nonlinear transformation to statisticalindependent quantities
� Mathematical methods:I Distributions as generalized functions (Analytization)I Laplace-transformI Comparison of Taylor-coefficients
� Advantages especially at high Reynolds-numbers (stability andadaptivity)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 57/60
Lattice Boltzmann methodsSummer term 2017
The highlights
� MRT-like-method with nonlinear transformation to statisticalindependent quantities
� Mathematical methods:I Distributions as generalized functions (Analytization)I Laplace-transformI Comparison of Taylor-coefficients
� Advantages especially at high Reynolds-numbers (stability andadaptivity)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 57/60
Lattice Boltzmann methodsSummer term 2017
The highlights
� MRT-like-method with nonlinear transformation to statisticalindependent quantities
� Mathematical methods:I Distributions as generalized functions (Analytization)I Laplace-transformI Comparison of Taylor-coefficients
� Advantages especially at high Reynolds-numbers (stability andadaptivity)
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 57/60
Lattice Boltzmann methodsSummer term 2017
Literature I
d’Humières, Dominique: Multiple–relaxation–time lattice Boltzmann models in three dimensions.Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and EngineeringSciences, 360(1792):437–451, 2002.
Dubois, François: Equivalent partial differential equations of a lattice Boltzmann scheme.Computers & Mathematics with Applications, 55(7):1441–1449, 2008.
Geier, M: Ab initio derivation of the cascaded Lattice Boltzmann Automaton.University of Freiburg–IMTEK, 2006.
Geier, Martin, Andreas Greiner und Jan G Korvink: Cascaded digital lattice Boltzmann automatafor high Reynolds number flow.Physical Review E, 73(6):066705, 2006.
Geier, Martin, Martin Schönherr, Andrea Pasquali und Manfred Krafczyk: The cumulantlattice Boltzmann equation in three dimensions: Theory and validation.Computers & Mathematics with Applications, 70(4):507–547, 2015.
Hänel, Dieter: Molekulare Gasdynamik: Einführung in die kinetische Theorie der Gase undLattice-Boltzmann-Methoden.Springer-Verlag, 2006.
Lukacs, Eugene: Characteristics functions.Griffin, London, 1970.
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 58/60
Lattice Boltzmann methodsSummer term 2017
Literature II
Ning, Yang und Kannan N Premnath: Numerical Study of the Properties of the Central MomentLattice Boltzmann Method.arXiv preprint arXiv:1202.6351, 2012.
Rohm, Florian: A commented Python Script for LBM-Flow-Simulations.Lecture on Lattice Boltzmann methods, 2015.
Wolf-Gladrow, Dieter A: Lattice-gas cellular automata and lattice Boltzmann models: AnIntroduction.Nummer 1725. Springer Science & Business Media, 2000.
Kumulanten-basierte LBM Prof. B. Wohlmuth Markus Muhr 59/60