BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
A new adaptative numerical method for kinetic
equation
Stéphane Brull, Louis Forestier-Coste, Luc Mieussens
MNMCFF2014 - Beijing
May 21-27, 2014
Forestier-Coste
local velocity grid
1/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Introduction
rarefied gas flow
Deterministic simulations (DVM) of Boltzmann equation
(BGK)
A global discrete velocity grid in space and time
For practical applications in aerodynamics (atmospheric
re-entry problems), the grid is so large that the computational
ressources (memory storage and CPU time) require by the
simulation are huge
Project : construction of local velocity grids (unsteady flows)
Forestier-Coste
local velocity grid
2/ 49
BGK Model
Standard method
Presentation of the method (1D)
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
5
Conclusions
Forestier-Coste
local velocity grid
2D method
Conclusions
3/ 49
BGK Model
Standard method
Presentation of the method (1D)
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
5
Conclusions
Forestier-Coste
local velocity grid
2D method
Conclusions
4/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
f (t, x, v ) distribution function : mass of particles at time t, at
position x ∈ Ω ⊂ RD with a velocity of v ∈ RD .
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
f (t, x, v ) distribution function : mass of particles at time t, at
position x ∈ Ω ⊂ RD with a velocity of v ∈ RD .
Z
mass density ρ(t, x) =
f (t, x, v )dv
RD
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
f (t, x, v ) distribution function : mass of particles at time t, at
position x ∈ Ω ⊂ RD with a velocity of v ∈ RD .
Z
mass density ρ(t, x) =
f (t, x, v )dv
RD
velocity
Forestier-Coste
u(t, x) =
1
ρ(t, x)
Z
v f (t, x, v )dv
RD
local velocity grid
5/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
f (t, x, v ) distribution function : mass of particles at time t, at
position x ∈ Ω ⊂ RD with a velocity of v ∈ RD .
Z
mass density ρ(t, x) =
f (t, x, v )dv
RD
velocity
u(t, x) =
1
ρ(t, x)
Z
energy
E (t, x) =
RD
Forestier-Coste
Z
v f (t, x, v )dv
RD
kv k2
f (t, x, v )dv
2
local velocity grid
5/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
f (t, x, v ) distribution function : mass of particles at time t, at
position x ∈ Ω ⊂ RD with a velocity of v ∈ RD .
Z
mass density ρ(t, x) =
f (t, x, v )dv
RD
velocity
1
u(t, x) =
ρ(t, x)
Z
energy
E (t, x) =
RD
temperature
Forestier-Coste
1
T (t, x) =
DR
Z
v f (t, x, v )dv
RD
kv k2
f (t, x, v )dv
2
2E (t, x)
− ku(t, x)k2
ρ(t, x)
local velocity grid
5/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
f (t, x, v ) distribution function : mass of particles at time t, at
position x ∈ Ω ⊂ RD with a velocity of v ∈ RD .
Z
mass density ρ(t, x) =
f (t, x, v )dv
RD
velocity
1
u(t, x) =
ρ(t, x)
Z
v f (t, x, v )dv
RD
Z
energy
temperature
Forestier-Coste
kv k2
f (t, x, v )dv
2
RD
Z
1
kv − u(t, x)k2 f (t, x, v )dv
T (t, x) =
DRρ(t, x) RD
E (t, x) =
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Notations
collisional invariants : m(v ) = 1, v , 21 kv k2
moments : ρ = (ρ, ρu, E )T
Z
< g >=
g dv
T
RD
ρ =< mf >
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
BGK model
∂t f + v .∇x f = Q(f )
Q(f ) =
1
(M[ρ,u,T ] − f )
τ
+ boundaries conditions
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Maxwellian
Maxwellian M[ρ,u,T ] (t, x, v ) : solution of an entropy minimisation
problem (H(f ) =< f ln(f ) >).
H(M[ρ,u,T ] ) = min{H(g ), g ≥ 0 s.t. < mg >= ρ}
M[ρ,u,T ] (t, x, v ) =
Forestier-Coste
ρ(t, x)
D
(2πRT (t, x)) 2
|v − u(t, x)|2
exp −
2RT (t, x)
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
5
Conclusions
Forestier-Coste
local velocity grid
2D method
Conclusions
9/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Discretization
Physical model:
∂t f + v · ∇x f = Q(f )
Velocity space RD
velocities
+ BCs
V = vk , k ∈ ND set of discrete
Discrete kinetic equation: ∂t fk + vk · ∇x fk = Qk (f )
Forestier-Coste
local velocity grid
+ BCs
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Velocity grid?
Cartesian
grid V: 0
8
1 0 1 0
19
l∆vx
l
0 : lmax =
<
V = vl,m,n = vmin + @m∆vy A , @mA = @0 : mmax A
:
;
n∆vz
n
0 : nmax
Constraints: bounds and grid step : the grid must be
large enough
fine enough
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
How to design a cartesian velocity grid?
p
2 RT (x)
Information on the support of f (x, .):
assuming f close to its local
Maxwellian, then centered
p on u(x)
with standard deviation RT (x)
p
u(x) − c RT (x)
u(x)
p
u(x) + c RT (x)
The grid contains all the distributions if the bounds are at least
p
p
vmin = min{u(x) − c RT (x)}
vmax = max{u(x) + c RT (x)}
x∈Ω
x∈Ω
(c around 4)
At least 3 points into the ”core” of each distribution. The grid step
should be:
p
∆v ≤ min{ 2RT (x)}
x∈Ω
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Estimation of the bounds and the step of the grid
Temperature and velocity fields are a priori unknown!
several solutions :
use the boundaries values: upstream flow (u∞ , T∞ ) and
velocity and temperature of the body (uwall , Twall )
Flow parameter inside the shock: ushock and Tshock estimated
by Rankine-Hugoniot relations.
Even better: A compressible Navier-Stokes pre-simulation :
u NS and T NS in Ω
let
vmin = min{u
x∈Ω
NS
(x) − c
p
RT NS (x)}
vmax = max{u NS (x) + c
x∈Ω
p
∆v = a min{ 2RT NS (x)}
p
RT NS (x)}
x∈Ω
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Drawback of a global grid
For reentry problem : the velocity is very large (≈ 6.000 m/s),
the temperature is very large in the shock (∼ 105 K), and very
small in the upstream and at the boundary (∼ 102 K)
Consequently: very large grid bounds,
and very small step, so very large
number of discrete velocities
Example: Mach 20, altitude 90 km: V contains 52 × 41 × 41
points. Around 350 GB memory requirements with a coarse
3D mesh in space!...
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Need of dynamics grids : example
two interacting blastwaves:
à t = 0
ρ = 1, u = 0, T = [1000, 0.01, 100]
√
max (u + 4 RT ) = 126
√
at t = 0.02 (after the interaction): max (u + 4 RT ) = 236:
larger bounds!
the optimal velocity grid get 2551 points!
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
Main Idea
Illustration
Extension of the LVG
4
2D method
5
Conclusions
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Local velocity grids
idea: define a velocity grid for each t and x
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Local velocity grid: problems
1
how to define the Local Velocity Grid: bounds and step?
2
how to exchange informations between two grids? Example:
∂x f (t, x, v ) ≈
f (t, x + ∆x, v ) − f (t, x, v )
∆x
but f (t, x, .) and f (t, x + ∆x, .) are not defined on the same
velocity grid...
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Definition of a LVG: conservation laws
at tn :
space cell [xi− 1 , xi+ 1 ]
2
2
n ≈ f (t , x , v n ), where
fi,k
n i i,k
n
o
n
n
n
n
vi,k
∈ Vin = vi,1
, vi,2
, . . . , vi,K
n
i
local velocity grid (with
Kin
points)
Uin ≈ U(tn , xi ) with a quadrature on the LVG:
Z
m(v )f (t n , xi , v ) dv = hmf (t n , xi , .)i
U(t n , xi ) = (ρ, ρu, E )(t n , xi ) =
R
↓
Kin
Uin = (ρni , ρni uin , Ein ) =
X
n
n n
m(vi,k
)fi,k
ωi,k = hmfi n iV n
i
k=1
where m(v ) = (1, v , 12 |v |2 ).
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Definition of a LVG: conservation laws
approximation of U(t n+1 , xi ) by discrete conservation laws:
conservation laws
∂t U + ∂x hvmf i = 0
finite volume upwind scheme
n
n
Uin+1 − Uin Φi+ 21 − Φi− 12
+
=0
∆t
∆x
where the numerical fluxes are defined by
n
Φni+ 1 = v + mfi n V n + v − mfi+1
Vn
2
Forestier-Coste
i
local velocity grid
i+1
20/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Definition of a LVG
approximations of u(tn+1 , xi ) and T (tn+1 , xi ): uin+1 and Tin+1
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Definition of a LVG
approximations of u(tn+1 , xi ) and T (tn+1 , xi ): uin+1 and Tin+1
bounds of Vin+1 set to
q
n+1
vmin,i
= uin+1 − 4 RTin+1
and
n+1
vmax,i
= uin+1 + 4
q
RTin+1
remark: corrects bounds if f close of its local Maxwellian
Forestier-Coste
local velocity grid
21/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Definition of a LVG
approximations of u(tn+1 , xi ) and T (tn+1 , xi ): uin+1 and Tin+1
bounds of Vin+1 set to
q
n+1
vmin,i
= uin+1 − 4 RTin+1
and
n+1
vmax,i
= uin+1 + 4
q
RTin+1
remark: corrects bounds if f close of its local Maxwellian
new discrete velocity grid Vin+1 : uniform with a constant
number of points (Kin+1 = 10 à 30)
remark: variables step and number of points could be
necessary (discontinuous f ).
Forestier-Coste
local velocity grid
21/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Computation of fi n+1
kinetic equation:
∂t f + v ∂x f = Q(f )
approximation of f (tn , xi , v ): fi n (v )
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Computation of fi n+1
kinetic equation:
∂t f + v ∂x f = Q(f )
approximation of f (tn , xi , v ): fi n (v )
finite volume - upwind scheme (continuous v ):
n
n
(v ) − fi+1
(v ) − fi n (v )
fi n+1 (v ) − fi n (v ) + fi n (v ) − fi−1
+v
+v
= Q(fi n )(v )
∆t
∆x
∆x
n and f n not defined on the same
problem: fi n+1 , fi n , fi−1
i+1
grids.
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Computation of fi n+1
reconstruction procedure: for each i, reconstruction of fi n on its LVG Vin
fin
(
f¯i n (v ) =
R(fi n )(v )
0
n
n
if vmin,i
≤ v ≤ vmax,i
f¯in (v)
else,
n
vmin,i
n
vi,k
n
vmax,i
numerical scheme:
n
f n (v ) − fi−1
(v )
fi n+1 (v ) − fi n (v )
f n (v ) − fi n (v )
+ v+ i
+ v − i+1
= Q(fi n )(v )
∆t
∆x
∆x
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Computation of fi n+1
reconstruction procedure: for each i, reconstruction of fi n on its LVG Vin
fin
(
f¯i n (v ) =
R(fi n )(v )
0
n
n
if vmin,i
≤ v ≤ vmax,i
f¯in (v)
else,
n
vmin,i
n
vi,k
n
vmax,i
numerical scheme:
n
f n (v ) − fi−1
(v )
fi n+1 (v ) − fi n (v )
f n (v ) − fi n (v )
+ v+ i
+ v − i+1
= Q(fi n )(v )
∆t
∆x
∆x
n+1
for every vi,k
of Vin+1 :
n+1
n+1
n+1
n+1
¯n n+1
¯n
¯n
¯n n+1
fi,k
− f¯i n (vi,k
)
n+1 + fi (vi,k ) − fi−1 (vi,k )
n+1 − fi+1 (vi,k ) − fi (vi,k )
+ vi,k
+ vi,k
∆t
∆x
∆x
n+1
n
¯
= Q(fi )(vi,k )
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Advantages of this scheme
1
advantage 1: the local velocity grid adapt in time and space
to the local temperature T and velocity u
2
advantage 2: only initial values for u and T are required
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Properties of the numerical scheme
1
if R positive, then the scheme is positive, provided that
∆t satisfies a standard CFL condition
Tin+1 positive
Forestier-Coste
local velocity grid
25/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Properties of the numerical scheme
1
if R positive, then the scheme is positive, provided that
∆t satisfies a standard CFL condition
Tin+1 positive
2
under this slight modification on the quadrature:
hfi n iV n := hf¯i n i
i
Tin+1 is positive
Forestier-Coste
local velocity grid
25/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Main Idea
Properties of the numerical scheme
1
if R positive, then the scheme is positive, provided that
∆t satisfies a standard CFL condition
Tin+1 positive
2
under this slight modification on the quadrature:
hfi n iV n := hf¯i n i
i
Tin+1 is positive
3
Forestier-Coste
in practice: even with non positive R, the scheme generally
preserves the positivity
local velocity grid
25/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
Main Idea
Illustration
Extension of the LVG
4
2D method
5
Conclusions
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Illustration 1: Sod test case
0,6
0,5
0,4
velocity profile:
0,3
0,2
0,1
0
0
0,1
0,2
0,3
0,4
0,5
... : global velocity grid (wrong bounds, with initial data)
- : global velocity grid (correct bounds, 100 points: bounds and step
estimated by an Euler computation, then a grid convergence study)
.- : local velocity grid (10 points)
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Illustration 2: two interracting blast waves
before the interaction (top), after (bottom)
T
u
3
p
20
400
2,5
15
300
2
10
1,5
200
5
1
100
0
0,5
0
0
0,2
0,4
0,6
0,8
1
4
-5
0
0,2
0,4
0,6
0,8
1
0
20
600
18
550
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
0,8
1
3
16
500
14
450
2
12
400
1
10
350
8
0
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
0,8
1
300
• global velocity grid: 2531 points (bounds and step estimated by an
Euler computation , then grid convergence study)
• local velocity grid: 30 points
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Reconstruction procedure
piecewise polynomial interpolation
high order interpolation is necessary
discontinuous distributions ( large Knudsen number), a non
oscillatory method is necessary (ENO interpolation used)
Forestier-Coste
local velocity grid
29/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Influence of the reconstruction step: Sod test
ρ
u
T
1
2
0,8
1,8
0,8
1,6
0,6
1,4
0,6
1,2
0,4
0,4
1
0,2
0,2
0,8
0,6
0
-1
-0,5
0
0,5
1
0
-1
-0,5
0
0,5
1
-1
-0,5
0
0,5
1
-0,5
0
0,5
1
3
1
0,8
2,5
0,8
0,6
2
0,6
1,5
0,4
0,4
1
0,2
0,2
0,5
0
-1
-0,5
0
0,5
1
0
-1
-0,5
0
0,5
1
0
-1
Free transport regime (no collisions):
top: global grid (30 points)
bottom: local velocity grid (30 points): affine interpolation,
ENO3, ENO4
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Boundaries conditions
−
diffuse reflection: f (t, x = 0, v > 0) = − hvhv+ Mfwi i Mw
ghost cell approach
Mw on its own local velocity grid (based on the wall
temperature and velocity)
however, such reflection can generate very discontinuous and
non-symmetric distributions!
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Boundaries conditions
example: Heat transfert
problem


ρ0
t = 0  u0 = 0 
T0 = TL
TL
x=0
x=0
Forestier-Coste
local velocity grid
TR
x=1
32/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Boundaries conditions
example: Heat transfert
problem


ρ0
t = 0  u0 = 0 
T0 = TL
TL
x=0
x=1
x=0
for small t, the support of
f (t, x ≈ 1, .) is non
symmetric:
Forestier-Coste
p
local velocity grid
TR
RTR
p
RTL
32/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Boundaries conditions
example: Heat transfert
problem


ρ0
t = 0  u0 = 0 
T0 = TL
TL
x=0
TR
x=0
x=1
for small t, the support of
f (t, x ≈ 1, .) is non
symmetric:
p
RTR
p
RTL
local velocity grid (symmetric by construction) is not large enough
for v < 0 (−1900 instead of −2200).
0
-10
-20
-30
-40
-50
0
0,1
0,2
0,3
0,4
0,5
0,6
(Kn = 0.01, global velocity grid=100 points, LVG=30 points)
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Extension of the LVG
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
Main Idea
Illustration
Extension of the LVG
4
2D method
5
Conclusions
Forestier-Coste
local velocity grid
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BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Extension of the LVG
Extension of the LVG: algorithm
n+1
n+1
compute fi,k
for every vi,k
of Vin+1
n+1
w = vi,1
(leftmost point)
loop
w = w − ∆vin+1
compute fi n+1 (w ) by the numerical scheme:
n
f¯n (w ) − f¯i−1
(w )
fi n+1 (w ) − f¯i n (w )
f¯n (w ) − f¯i n (w )
+w + i
+w − i+1
= Q̄(fi n )(w )
∆t
∆x
∆x
if fi n+1 (w ) is too large then add w to the grid and continue
the loop
else stop
Forestier-Coste
local velocity grid
34/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Extension of the LVG
Extension of the LVG: illustration
Heat transfert problem:
T
1000
u
p
3
0
900
-10
2,5
800
-20
700
2
600
-30
500
1,5
-40
400
300
0
0,1
0,2
0,3
0,4
0,5
0,6
900
-50
0
0,1
0,2
0,3
0,4
0,5
0,6
1
0
0,1
0,2
0,3
0,4
0,5
0
0,1
0,2
0,3
0,4
0,5
0,6
3
0
800
-10
2,5
700
-20
600
2
-30
500
1,5
-40
400
300
0
0,1
0,2
0,3
0,4
0,5
0,6
-50
0
0,1
0,2
0,3
0,4
0,5
0,6
1
no extension, symmetric LVG (top)
with extension (bottom)
Kn = 0.01, global velocity grid=100 points, LVG=30 points
Forestier-Coste
local velocity grid
35/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
2D preliminary results
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
2D preliminary results
Embedded local velocity grid
Illustration
Mesh Refinment
5
Conclusions
Forestier-Coste
local velocity grid
36/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
2D preliminary results
To the 2D
reduct model : F (t, x, y , vx , vy , vz )
f (t, x, y , vx , vy ) = hF iz
g (t, x, y , vx , vy ) = hF
|vz |2
iz
2
cartesian velocity grid
reconstruction : 2D ENO interpolation
Forestier-Coste
local velocity grid
37/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
2D preliminary results
Heat transfert problem on a cylinder:
outside wall temperature : 1000K
inside wall and domain temperature : 300K
gas at equilibrium
Forestier-Coste
local velocity grid
38/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
2D preliminary results
Heat transfert problem on a cylinder:
outside wall temperature : 1000K
inside wall and domain temperature : 300K
gas at equilibrium
Forestier-Coste
local velocity grid
38/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
2D preliminary results
computational time too long
Each velocity of each cell interpolated at least 8 times.
Forestier-Coste
local velocity grid
39/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
2D preliminary results
computational time too long
Each velocity of each cell interpolated at least 8 times.
⇒ new definition of local velocity grids
Forestier-Coste
local velocity grid
39/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Embedded local velocity grid
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
2D preliminary results
Embedded local velocity grid
Illustration
Mesh Refinment
5
Conclusions
Forestier-Coste
local velocity grid
40/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Embedded local velocity grid
Embedded LVG : definiton
√
u ± 4 RT
reference velocity step.
Forestier-Coste
local velocity grid
41/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Embedded local velocity grid
Embedded LVG : definiton
√
u ± 4 RT
reference velocity step.
reference point for the grids : (0, 0).
Forestier-Coste
local velocity grid
41/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Embedded local velocity grid
Embedded LVG : definiton
√
u ± 4 RT
reference velocity step.
reference point for the grids : (0, 0).
take into account the accuracy wished.
Forestier-Coste
local velocity grid
41/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Embedded local velocity grid
Need of auto extension
cellule 1130
-43.4373
cellule 1130
-12.412
-41.4344
-12.4799
2954.9
2482.1
v_y
v_y
-2482.1
-2009.3
3427.7
-2954.9
v_x
-2363.9
without
Forestier-Coste
v_x
3427.7
with
local velocity grid
42/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
2D preliminary results
Embedded local velocity grid
Illustration
Mesh Refinment
5
Conclusions
Forestier-Coste
local velocity grid
43/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Shock wave on a cylinder
high velocity wave coming
from left.
Knudsen number around 1.
Forestier-Coste
local velocity grid
44/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
T
2567
2000
1000
236
Forestier-Coste
local velocity grid
45/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
Error
0.057
0.04
0.02
0
Forestier-Coste
local velocity grid
45/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Illustration
N
63
60
50
40
30
17
Forestier-Coste
local velocity grid
20
45/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Mesh Refinment
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
2D preliminary results
Embedded local velocity grid
Illustration
Mesh Refinment
5
Conclusions
Forestier-Coste
local velocity grid
46/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Mesh Refinment
If extrem rarefied then far from equilibrium.
Distributions can be multimodal.
last method : good boundaries but lot of points to see the
thinest mode
AMR idea :
take the boundaries from last method.
refine only where the mode are thin.
Forestier-Coste
local velocity grid
47/ 49
BGK Model
Standard method
Presentation of the method (1D)
1
BGK Model
2
Standard method
3
Presentation of the method (1D)
4
2D method
5
Conclusions
Forestier-Coste
local velocity grid
2D method
Conclusions
48/ 49
BGK Model
Standard method
Presentation of the method (1D)
2D method
Conclusions
Conclusions
smaller velocity grid
time-consuming high order ENO interpolation
Embedded LVG
Perspectives
better extension of 2D LVG
3D
Forestier-Coste
local velocity grid
49/ 49