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The Lattice Boltzmann methodfor hyperbolic systems
Benjamin GrailleOctober 19, 2016
Framework
The Lattice Boltzmann method
1 Description of the lattice Boltzmann methodLink with the kinetic theoryClassical schemesAnalysis methodsBoundary conditions
2 pyLBM (collaboration with Loïc Gouarin)PresentationExamples
3 The vectorial schemesPresentationNumerical tests
4 Conclusion
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Description of the lattice Boltzmann method
The Lattice Boltzmann method
1 Description of the lattice Boltzmann methodLink with the kinetic theoryClassical schemesAnalysis methodsBoundary conditions
2 pyLBM (collaboration with Loïc Gouarin)
3 The vectorial schemes
4 Conclusion
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Description of the lattice Boltzmann method – Link with the kinetic theory
Different scales for modeling
microscopicmodels
macroscopicmodels
mesoscopicmodels
latticeBoltzmann
Chapman-Enskogε→ 0
Taylor expansion∆t → 0
particlespositions, velocities
statistical descriptionsmean distribution functions
observable quantitiesdensity, velocity, temperature
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Description of the lattice Boltzmann method – Link with the kinetic theory
From mesoscopic to macroscopic
Mesoscopic scale: the Boltzmann equation
∂t f (t , x , c) + c·∇x f (t , x , c) =1
εQ(f ),
where ε is the Knudsen number.Chapman-Enskog method: formal asymptotic expansion according to ε⇒ the macroscopic equations on the moments.
ρ =
∫f (t , x , c) dc, (mass)
q =
∫cf (t , x , c) dc, (momentum)
E =
∫12c2f (t , x , c) dc, (energy)
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Description of the lattice Boltzmann method – Link with the kinetic theory
Mimic the kinetic to simulate the macroscopicExample in 2 dimensions
a uniform cartesian mesh
on each spot, “particles” with adapted discret velocitiesthe transport phasethe relaxation phasedo it again !
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Description of the lattice Boltzmann method – Link with the kinetic theory
Mimic the kinetic to simulate the macroscopicExample in 2 dimensions
a uniform cartesian meshon each spot, “particles” with adapted discret velocities
the transport phasethe relaxation phasedo it again !
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Description of the lattice Boltzmann method – Link with the kinetic theory
Mimic the kinetic to simulate the macroscopicExample in 2 dimensions
a uniform cartesian meshon each spot, “particles” with adapted discret velocitiesthe transport phase
the relaxation phasedo it again !
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Description of the lattice Boltzmann method – Link with the kinetic theory
Mimic the kinetic to simulate the macroscopicExample in 2 dimensions
a uniform cartesian meshon each spot, “particles” with adapted discret velocitiesthe transport phasethe relaxation phase
do it again !
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Description of the lattice Boltzmann method – Link with the kinetic theory
Mimic the kinetic to simulate the macroscopicExample in 2 dimensions
a uniform cartesian meshon each spot, “particles” with adapted discret velocitiesthe transport phasethe relaxation phasedo it again !
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Description of the lattice Boltzmann method – Link with the kinetic theory
Sketch of the collision phase
even nonconservedmoments
odd nonconservedmoments
conservedmoments
s > 1
s < 1
f
f eqf ?
The collision operator is a linear re-laxation of the vector f toward anequilibrium value f eq
f ? = f + M−1SM (f eq − f )).
The value of f eq isfunction of the con-served moments.
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Description of the lattice Boltzmann method – Link with the kinetic theory
Sketch of the collision phase
even nonconservedmoments
odd nonconservedmoments
conservedmoments
s > 1
s < 1
f
f eqf ?
The collision operator is a linear re-laxation of the vector f toward anequilibrium value f eq
f ? = f + M−1SM (f eq − f )).
The value of f eq isfunction of the con-served moments.
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Description of the lattice Boltzmann method – Link with the kinetic theory
Define a lattice Boltzmann schemeA lattice Boltzmann scheme is given by
a set of q velocities adapted to the mesh, c0, . . . , cq−1,an invertible matrixM that transforms the densities into the moments:
mk =
q−1∑j=0
Mkj fj =
q−1∑j=0
Pk (cj )fj , Pk ∈ R[X ],
functions defining the equilibrium meqk ,
relaxation parameters sk .
The nth first moments are necessary the unknowns of the PDEs: thesemoments are conserved, that is
meqk = mk , 0 6 k 6 n − 1.
The next moments are not conserved and their equilibrium value dependson the conserved moments:
meqk = meq
k (m0, . . . ,mn−1), n 6 k 6 q − 1.
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Description of the lattice Boltzmann method – Classical schemes
Examples
D1Q2
advection, heat
0 12
D1Q3
advection, wave, heat
0 12
D1Q5
Euler
0 12 34
D2Q4
advection, heat
0 1
2
3
4
D2Q5
advection, wave, heat
0 1
2
3
4
D2Q9
Navier-Stokes
0 1
2
3
4
56
7 8
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Description of the lattice Boltzmann method – Analysis methods
Example of the D1Q2
The D1Q2 can be used to simulate the mono-dimensional hyperbolicconservative equation
∂tu(t , x) + ∂xϕ(u)(t , x) = 0, t > 0, x ∈ R,
where u : R→ R is the unknown.The D1Q2 is given by
two velocities {−λ, λ} with λ = ∆x/∆t and the associated densities (f− , f+ )
the matrixM and it’s inverse
M =
(1 1−λ λ
)M−1 =
(1/2 −1/(2λ)1/2 1/(2λ)
)the conserved moment u and the non conserved moment v, where u = f− + f+and v = −λf− + λf+ .
the equilibrium value veq = veq(u) and the relaxation parameter s .
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Description of the lattice Boltzmann method – Analysis methods
One time step of the linear D1Q2
In the linear case (veq = αu), one time step of the scheme reads
v?(x , t) = (1− s)v(x , t) + sαu(x , t), relaxation,f− (x , t + ∆t) = f ?− (x + ∆x , t), transport to the left,f+ (x , t + ∆t) = f ?+ (x −∆x , t), transport to the right.
In terms of densities, it reads
f− (x , t + ∆t) = 12(2− s − sα
λ)f− (x + ∆x , t) + 1
2(s − sα
λ)f+ (x + ∆x , t),
f+ (x , t + ∆t) = 12(s + sα
λ)f− (x + ∆x , t).+ 1
2(2− s + sα
λ)f+ (x + ∆x , t)
In terms of moments, it reads
un+1j = 1
2(1− s α
λ)unj+1 + 1
2(1 + s α
λ)unj−1 − 1−s
2λ(vnj+1 − vnj−1),
vn+1j = 1−s
2(vnj+1 + vnj−1)− λ
2(1− s α
λ)unj+1 + λ
2(1 + s α
λ)unj−1.
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Description of the lattice Boltzmann method – Analysis methods
Equivalent equations (0)
Assuming that the densities are regular functions, the Taylor expansionmethod for small ∆t and ∆x yields
Zeroth order
At the order 0, the distribution are at equilibrium
fj = f eqj + O(∆t ), f ?j = f eqj + O(∆t ), j ∈ {−,+}.
We have
fj (t + ∆t , x) = f ?j (t , x − vj∆t), vj = jλ, j ∈ {−,+},fj + O(∆t ) = f ?j + O(∆t ),
v = v? + O(∆t ) = (1− s)v+ sveq + O(∆t ),
v = veq + O(∆t ) and v? = veq + O(∆t ).
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Description of the lattice Boltzmann method – Analysis methods
Equivalent equations (1)
First orderThe first moment u satisfies the partialdifferential equation
∂tu+ ∂x veq = O(∆t ).
The choice veq = ϕ(u) is then done sothat u satisfies the conservative equa-tion at order 1.
Transition lemmaThe second moment v satisfies
v = veq − ∆tsθ + O(∆t2),
v? = veq + ∆t(1− 1
s
)θ + O(∆t2),
withθ = ∂tv
eq + λ2∂xu.
We have
fj (t + ∆t , x) = f?j (t , x − vj ∆t),
fj + ∆t∂t fj = f?j − vj ∆t∂x f
?j + O(∆t2),
fj + ∆t∂t feqj = f
?j − vj ∆t∂x f
eqj + O(∆t2).
Taking the first moment:
u+ ∆t∂tu = u−∆t∂x veq + O(∆t2).
Taking the second moment:
v+ ∆t∂tveq = v? − λ2∆t∂xu+ O(∆t2)
= (1− s)v+ sveq
− λ2∆t∂xu+ O(∆t2)
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Description of the lattice Boltzmann method – Analysis methods
Equivalent equations (2)
Second order
The first moment u satisfies the second-order partial differentialequation
∂tu+ ∂xϕ(u) = ∆t σ ∂x[(λ2 − ϕ′(u)2)∂xu]+ O(∆t2),
with σ = 1/s − 1/2.
Taking the first moment at second order:
u+ ∆t∂tu+ 12∆t2∂ttu = u−∆t∂xv? + 1
2λ2∆t2∂xxu+ O(∆t3)
∂tu+ ∂xϕ(u) = ∂x (veq − v?) + 12∆t(λ2∂xxu− ∂ttu) + O(∆t2)
veq − v? = ∆t( 1s− 1)θ θ = ϕ′(u)∂tu+ λ2∂xxu = (λ2 − ϕ′(u)2)∂xxu
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Description of the lattice Boltzmann method – Analysis methods
Maximum principle
Theorem
We assume that s ∈ [0, 1], u0j ∈ [α, β], v0
j = ϕ(u0j ), ∀j ,
and λ > maxα6u6β
|ϕ′(u)|.Then ∀n > 0 ∀j unj ∈ [α, β].
The functions h±(u) = λu±ϕ(u)2λ
are increasing. We then have
f 0±,j ∈ [h±(α), h±(β)].
The relaxation phase reads as a linear convex combination:
f n?±,j =λunj ± vn?j
2λ=λunj ± vnj
2λ±vn?j − vnj
2λ= f n±,j ± s
ϕ(unj )− vnj2λ
= (1− s)f n±,j + sh±(unj ) ∈ [h±(α), h±(β)].
And
unj = f n−,j + f n+,j ∈ [h−(α) + h+(α), h−(β) + h+(β)] = [α, β].
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Description of the lattice Boltzmann method – Analysis methods
L2 stability
Theorem
We assume that veq = αu, λ > |α|, and s ∈ [0, 2].Then the D1Q2 scheme is stable for the L2-norm.
The amplification matrix of the linear D1Q2 scheme is given by
G(∆x , ξ) =
((1− s
2(1 + α
λ))e−i∆xξ s
2(1− α
λ)e−i∆xξ
s2(1 + α
λ)e i∆xξ
(1− s
2(1− α
λ))e i∆xξ
).
real part
1.00.5
0.00.5
1.0im
aginary p
art
1.0
0.5
0.0
0.5
1.0
s
0.0
0.5
1.0
1.5
2.0
Eigenvalues for α= 0. 25
real part
1.00.5
0.00.5
1.0im
aginary p
art
1.0
0.5
0.0
0.5
1.0
s
0.0
0.5
1.0
1.5
2.0
Eigenvalues for α= 0. 5
real part
1.00.5
0.00.5
1.0im
aginary p
art
1.0
0.5
0.0
0.5
1.0
s
0.0
0.5
1.0
1.5
2.0
Eigenvalues for α= 0. 75
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Description of the lattice Boltzmann method – Boundary conditions
Bounce Back and anti Bounce Back
�
�
�
�
�
�
�
�
�
The black point has outside neigh-bors.For each one, we have to specify thedistribution functions with incomingvelocities.
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Description of the lattice Boltzmann method – Boundary conditions
Bounce Back and anti Bounce Back
�
�
�
�
�
�
�
�
�
◦
◦
◦ When a fluid particle (discrete distri-bution function) reaches a boundarynode, the particle will scatter back tothe fluid along with its incoming direc-tion.For anti Bounce Back condition, thesign is changed.The bound of the domain is then pix-elized.
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Description of the lattice Boltzmann method – Boundary conditions
Bouzidi type boundary conditions
In order to improve the accuracy of the boundary conditions, the interfaceneed to be localized more precisely.
s < 1/2� � �
xi−1 xi xi+1
◦
s
1−s
s > 1/2� � �
xi−1 xi xi+1
◦
s
1−s
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Description of the lattice Boltzmann method – Boundary conditions
Bouzidi type boundary conditions
Case s < 1/2 (explicit interpolation).
s < 1/2� � �
xi−1 xi xi+1xi− 1
2
◦
s
1−s 1−s
We havexi− 1
2= (1− 2s)xi−1 + 2sxi .
Thenf −i+1 = f̃ −i = f +
i− 12
= (1− 2s)f +i−1 + 2sf +
i .
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Description of the lattice Boltzmann method – Boundary conditions
Bouzidi type boundary conditions
Case s > 1/2 (implicit interpolation).
s > 1/2� � �
xi−1 xi xi+1xi+1
2
◦
s
1−s 1−s
We havexi =
1
2sxi+ 1
2+
2s − 1
2sxi−1.
Thenf −i+1 = f̃ −i =
1
2sf̃ −i+ 1
2+
2s − 1
2sf̃ −i−1 =
1
2sf +i +
2s − 1
2sf −i
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pyLBM (collaboration with Loïc Gouarin)
The Lattice Boltzmann method
1 Description of the lattice Boltzmann method
2 pyLBM (collaboration with Loïc Gouarin)PresentationExamples
3 The vectorial schemes
4 Conclusion
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pyLBM (collaboration with Loïc Gouarin) – Presentation
Motivations
pyLBMacademic and flexible code
to test and to compare the schemes: simple and brief implementationcan be used by non specialists or students
optimisations: mpi, openmp, cython (cuda and opencl in test)⇒ python: compromise efficiency / simplicity
fine management of the geometry built from simple objects2D: circle, ellipse, triangle, parallelogram3D: sphere, ellipsoid, parallelepiped, cylindrer
The user definesthe geometry of the domain and the scheme through a dictionary,initial conditions and boundary conditions through functions,parameters for the optimization (optional),
the code is then generated and executed.
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pyLBM (collaboration with Loïc Gouarin) – Presentation
Structure of the code
Dictionnary
Simulation
Geometry
Circle
Ellipse
Parallelogram
Triangle
Sphere
Ellipsoid
Cylinder
Parallelepiped
Stencil Velocity
Domain Scheme Generator
Array : f , m Fonctions :transportrelaxationm2f f2m
Result
Initialization
Boundary conditions
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pyLBM (collaboration with Loïc Gouarin) – Examples
D1Q2 for the advection
Geometryimport pyLBM
xmin , xmax = 0., 1. # domain
dic = {' box ' :{ 'x ': [xmin , xmax]},
}
geom = pyLBM.Geometry(dic)print(geom)geom.visualize ()
Geometry informations
spatial dimension: 1
bounds of the box:
[[ 0. 1.]]
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pyLBM (collaboration with Loïc Gouarin) – Examples
D1Q2 for the advection
Stencilimport pyLBM
dic = {' dim ': 1,' schemes ' :[
{' velocities ': [1, 2],
}],
}
sten = pyLBM.Stencil(dic)print(sten)sten.visualize ()
Stencil informations
* spatial dimension: 1
* maximal velocity in each direction: [1]
* minimal velocity in each direction: [-1]
* Informations for each elementary stencil:
stencil 0
- number of velocities: 2
- velocities: (1: 1), (2: -1),
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pyLBM (collaboration with Loïc Gouarin) – Examples
D1Q2 for the advection
Domainimport pyLBM
xmin , xmax = 0., 1. # domainN = 10 # number of points
dic = {' box ' :{ 'x ': [xmin , xmax]},' space_step ': (xmax -xmin)/N,' scheme_velocity ': 1.,' schemes ' :[
{' velocities ': [1, 2],
}],
}
dom = pyLBM.Domain(dic)print(dom)dom.visualize ()
Domain informations
spatial dimension: 1
space step: dx= 1.000e-01
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pyLBM (collaboration with Loïc Gouarin) – Examples
D1Q2 for the advection
Schemeimport pyLBMimport sympy as sp
u, X = sp.symbols( 'u, X ')c = .25 # velocitys = 1. # relaxation parameterdic = {
' dim ': 1,' scheme_velocity ': 1.,' schemes ' :[
{' velocities ': [1, 2],' conserved_moments ': u,' polynomials ': [1, X],' equilibrium ': [u, c*u],' relaxation_parameters ': [0., s],' init ': {u: 0.}
}],
}scheme = pyLBM.Scheme(dic)print(scheme)
Scheme informations
spatial dimension: dim=1
number of schemes: nscheme=1
number of velocities:
Stencil.nv[0]=2
velocities value:
v[0]=(1: 1), (2: -1),
polynomials:
P[0]=Matrix([[1], [X]])
equilibria:
EQ[0]=Matrix([[u], [0.25*u]])
relaxation parameters:
s[0]=[0.0, 1.0]
moments matrices
M = [Matrix([
[1, 1],
[1, -1]])]
invM = [Matrix([
[1/2, 1/2],
[1/2, -1/2]])]
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pyLBM (collaboration with Loïc Gouarin) – Examples
D1Q2 for the advection (numerical results)Sm
oothsolution
k s 2.000 1.900 1.750 1.000 0.750 0.500
3 1.536e-01 1.416e-01 1.256e-01 8.104e-02 7.881e-02 8.113e-02
4 1.733e-01 1.714e-01 1.712e-01 2.062e-01 2.288e-01 2.550e-01
5 1.319e-01 1.153e-01 1.073e-01 1.495e-01 1.757e-01 2.100e-01
6 4.897e-02 4.697e-02 5.138e-02 1.145e-01 1.405e-01 1.719e-01
7 1.254e-02 1.429e-02 2.162e-02 7.983e-02 1.049e-01 1.357e-01
8 3.113e-03 4.850e-03 9.913e-03 4.990e-02 7.081e-02 9.927e-02
9 7.761e-04 1.991e-03 4.836e-03 2.863e-02 4.329e-02 6.599e-02
10 1.943e-04 9.263e-04 2.412e-03 1.551e-02 2.448e-02 3.990e-02
11 4.863e-05 4.522e-04 1.208e-03 8.096e-03 1.311e-02 2.233e-02
12 1.216e-05 2.241e-04 6.041e-04 4.138e-03 6.794e-03 1.188e-02
13 3.039e-06 1.117e-04 3.022e-04 2.092e-03 3.461e-03 6.136e-03
14 7.598e-07 5.577e-05 1.512e-04 1.052e-03 1.747e-03 3.121e-03
15 1.900e-07 2.787e-05 7.559e-05 5.277e-04 8.778e-04 1.574e-03
16 4.749e-08 1.393e-05 3.780e-05 2.642e-04 4.400e-04 7.904e-04
slope 2.000e00 1.000e00 9.999e-01 9.979e-01 9.965e-01 9.937e-01
Discontinuoussolution k s 2.000 1.900 1.750 1.000 0.750 0.500
3 2.722e-01 2.657e-01 2.590e-01 2.649e-01 2.758e-01 2.893e-01
4 8.353e-02 8.611e-02 9.415e-02 1.696e-01 2.027e-01 2.389e-01
5 1.488e-01 1.372e-01 1.304e-01 1.434e-01 1.587e-01 1.832e-01
6 1.055e-01 9.036e-02 8.323e-02 1.066e-01 1.225e-01 1.444e-01
7 8.651e-02 7.416e-02 7.188e-02 9.591e-02 1.082e-01 1.251e-01
8 6.158e-02 4.995e-02 5.070e-02 7.838e-02 8.932e-02 1.038e-01
9 5.568e-02 4.470e-02 4.497e-02 6.609e-02 7.494e-02 8.675e-02
10 4.421e-02 3.434e-02 3.570e-02 5.515e-02 6.270e-02 7.268e-02
11 3.460e-02 2.684e-02 2.954e-02 4.657e-02 5.289e-02 6.125e-02
12 2.710e-02 2.089e-02 2.424e-02 3.909e-02 4.442e-02 5.146e-02
13 2.230e-02 1.732e-02 2.043e-02 3.288e-02 3.735e-02 4.326e-02
14 1.783e-02 1.406e-02 1.707e-02 2.763e-02 3.140e-02 3.637e-02
15 1.403e-02 1.151e-02 1.432e-02 2.324e-02 2.641e-02 3.059e-02
16 1.111e-02 9.517e-03 1.202e-02 1.954e-02 2.220e-02 2.572e-02
slope 3.374e-01 2.746e-01 2.527e-01 2.502e-01 2.501e-01 2.501e-01
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The vectorial schemes
The Lattice Boltzmann method
1 Description of the lattice Boltzmann method
2 pyLBM (collaboration with Loïc Gouarin)
3 The vectorial schemesPresentationNumerical tests
4 Conclusion
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The vectorial schemes – Presentation
Motivations
We focus on the numerical simulation of hyperbolic systems of pde’s by thelattice Boltzmann method. In this presentation, we deal with monodimensional conservation laws:
∂tu(t , x) + ∂xϕ(u)(t , x) = 0, t > 0, x ∈ R,
where u is the unknown vector of size n .
Goals:generic scheme for all flux functions ϕ,identifiable stability conditions,straightforward treatment of the boundary conditions.
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The vectorial schemes – Presentation
The vectorial D1Q2,...,2
For a system of n equations, we duplicate n times the D1Q2:the velocity of the scheme: λ = ∆x/∆t
the velocities of the particles: v0 = −λ, v1 = λ
the particles distributions: f1,0 , f1,1 , . . . , fn,0 , fn,1the moments: uk = fk,0 + fk,1 , vk = λ(−fk,0 + fk,1 ), 1 6 k 6 n
the relaxation phase: u?k = uk , v?k = vk + sk (veqk − vk ), 1 6 k 6 n
Theorem (Equivalent equation)
Taking veqk = ϕk (u), the vector of the first moments u satisfy
∂tu+ ∂xϕ(u) = ∆t S ∂x
((λ2In −
(dϕ(u)
)2)∂xu)
+ O(∆t2),
with S = diag(σ1, . . . , σn), σk = 1/sk − 1/2, 1 6 k 6 n .
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The vectorial schemes – Presentation
The D1Q2,...,2 as a relaxation schemeThe relaxation system proposed by Jin and Xin reads
{∂tu
ε + ∂xvε = 0,
∂tvε + A∂xuε = 1
ε(ϕ(uε)− vε).
Denoting uni = u(xi , tn), vni = v(xi , tn), xi ∈ L, tn = n∆t , the D1Q2,...,2 can
be rewritten in the form (if all relaxation parameters are equal to s)
vn?i = vni − s(vni − ϕ(uni )),
un+1i = 1
2(uni+1 + uni−1)− ∆t
2∆x(vn?i+1 − vn?i−1),
vn+1i = 1
2(vn?i+1 + vn?i−1)− λ2 ∆t
2∆x(uni+1 − uni−1).
Splitting betweenthe relaxation part treated by the explicit Euler method with ε = ∆t/s ,the hyperbolic part treated by the Lax-Friedrichs method with A = λ2In .
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The vectorial schemes – Numerical tests
Numerical tests done with pyLBM
dimension 1advection with constant velocityadvection-diffusion-reaction equationBurgersp-systemshallow water systemfull compressible Euler system
dimension 2advection with constant velocityshallow water systemincompressible thermo-hydrodynamic (Boussinesq approximation)magneto-hydro-dynamic system
dimension 3advection with constant velocityincompressible thermo-hydrodynamic (Boussinesq approximation)
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Conclusion
The Lattice Boltzmann method
1 Description of the lattice Boltzmann method
2 pyLBM (collaboration with Loïc Gouarin)
3 The vectorial schemes
4 Conclusion
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Conclusion
The Lattice Boltzmann method
The lattice Boltzmann method and the Boltzmann equationvery rough discretization in term of velocitiesno claim to solve the Boltzmann equationa taylor expansion to mimic the Chapman-Enskog expansion
The software pyLBMflexible and simple interface to create a simulationperformances of a compiled langagesome features to add: optimization with grpahics cards, relative velocity schemes(Tony Février), mesh refinement...
The vectorial schemessimple way to simulate conservative hyperbolic problemslink with the relaxation methodsto do: obtain a convergence theorem with this restrictive framework
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