The Lattice Boltzmann method - for hyperbolic systems · Framework The Lattice Boltzmann method 1 Description of the lattice Boltzmann method Link with the kinetic theory Classical

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

2 / 36GRAILLE / NUMKIN2016

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Description of 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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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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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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a uniform cartesian meshon each spot, “particles” with adapted discret velocities

the transport phasethe relaxation phasedo it again !


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a uniform cartesian meshon each spot, “particles” with adapted discret velocitiesthe transport phase

the relaxation phasedo it again !


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a uniform cartesian meshon each spot, “particles” with adapted discret velocitiesthe transport phasethe relaxation phase

do it again !


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a uniform cartesian meshon each spot, “particles” with adapted discret velocitiesthe transport phasethe relaxation phasedo it again !


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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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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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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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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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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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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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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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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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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


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



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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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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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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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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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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)


1 Description of the lattice Boltzmann method

2 pyLBM (collaboration with Loïc Gouarin)PresentationExamples


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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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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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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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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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




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.

31 / 36GRAILLE / NUMKIN2016N


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 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





4 Conclusion


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Conclusion


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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Documents

The Lattice Boltzmann method - for hyperbolic systems · Framework The Lattice Boltzmann method 1 Description of the lattice Boltzmann method Link with the kinetic theory Classical