A Multiscale Approach to the Boltzmann-BGK Equation
1
2
3
Alan Medinger , Rachel Nevin , Emma Talis , and Jae-Jae Young
4
2015 Iowa State University REU in Mathematics5
Operator Splitting and Semi-Lagrangian Time-Stepping
Motivation
Our goal is to develop a multiscale method that combines the specificity of a
kinetic solver with the efficiency of a fluid solver.
I This multiscale solver should be asymptotic-preserving method:
I
∆x, ∆v, ∆t → 0
Numerical Method
f,t + vf,x = 1ε (M − f )
1 ∆tB ∆tA 1 ∆tB
e
e2
e2
I
+
Second-order Strang operator splitting, with A = Advection Operator
and B = Collision Operator:
I
Rewriting collision step:
I
I
1 ∆t time-step on f = 1 (M − f )
,t
ε
2
I
∆t time-step on f,t + vf,x = 0
Restricting to a single cell in 2D:
⇒
Numerical Method
for Compressible
Euler Equations
∆x, ∆v, ∆t → 0+
Projecting initial condition onto a 2D mesh:
f (x, v)
Compressible
Euler Equations
The kinetic model considered in this work is the 1D1V Boltzmann equation with
the Bhatnagar-Gross-Krook (BGK) collision operator:
1
f,t + vf,x = (M − f )
ε
3
ρ2
M(t, x, v) = √
where
2πp
−ρ(u−v)2
e 2p .


I
where
1 2 1
E = ρu + p,
2
2
k = 1, 2, ..., M : index of 1D Legendre coefficient
M (M +1)
I ` = 1, 2, ...,
: index of 2D Legendre coefficient
2
∗
I η : constant; Gaussian quadrature point in Tij denoting the 1D line onto
which we project our function
v∆t
t = ∆t
u := macroscopic velocity,
p := pressure,
and E := energy,
1
and E =
2
i
i+1
∆x
Figure 1: A visualization of a function shifting across a fixed mesh in time.
Fluid Solver Results: Shock Tube Problem
Z
v 2f dv.
I
In this work we couple a semi-Lagrangian kinetic solver and a Runge-Kutta fluid
solver to produce a multiscale method.
Moments of probability density function (PDF):
Z ∞
Mγ =
v γ f dv
rarefaction


ρ`
 u` 
t↑
p`
−∞
where
M0 = ρ,
Discontinuous Galerkin Method
I
(1-Dimensional)
I
∆x
Dividing our domain into a mesh of
equally sized cells with x ∈ [a, b]:
Numerically approximated solution:
∆x
f (t, x)
x 1 ,x 1
i− 2 i+ 2
a
b
M1 = ρu,
M2 = 2E
Shock tube initial condition with discontinuity at x = 0:

  
   
ρr
0.125
1
ρ`
left state:  u`  =  0  right state:  ur  =  0 
p`
0.1
1
pr
contact
 ∗   ∗ 
ρ`
ρr
 u∗   u∗  shock


p∗
p∗
ρr
 ur 
pr
I
I
2
ξ=
(x − xi)
∆x
I
Numerically approximated solution:
∆x
f (t, x, v)
p = M2 −
where
∆t time-step:
I From fluid solution:
2
−ρn+1 n+1
u
−v
n+1
e 2p
I
where
1 2 1
E = ρu + p
2
2
Using Strang splitting, solve
with frozen M :
1
f,t + vf,x = (M − f )
ε
[1] M. Bennoune, M. Lemou, L. Mieussens. Uniformly stable numerical schemes for
the Boltzmann equation preserving the compressible Navier–Stokes asymptotics. J.
Comp. Phys., 227 (2008):3781-3803.
[2] P. L. Bhatnagar, E. P. Gross, M. Krook, A model for collision processes in gases
I. Small amplitude processes in charged and neutral one-component systems. Phys.
Rev.. 94 (1954): 511-525.
[3] B. Cockburn, C. Shu, The Runge-Kutta discontinuous Galerkin method for
conservation laws V. J. Comp. Phys.. 141 (1998): 199-224.
[4] G. Strang, On the construction and comparison of difference schemes. SIAM J.
Numer. Anal.. 5 (1968): 506-517.
[5] J. A. Rossmanith, D. C. Seal, A positivity-preserving high-order semi-Lagrangian
discontinuous Galerkin scheme for the Vlasov-Poisson equations. J. Comp. Phys..
230 (2011): 6203-6232.
[6] G. A. Sod, A survey of several finite difference methods for systems of nonlinear
hyperbolic conservation laws. J. Comp. Phys.. 27. (1978): 1-31.
[7] S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic
equations: a review. Rivista Di Matematica Della Università Di Parma. 3
(2012): 177-216.
[8] L. Krivodonova, Limiters for high-order discontinuous Galerkin methods. J.
Comp. Phys.. 226 (2007): 879-896.
M12
M0 .
b
I
(`)
Fij (t) ψ (`)(ξ, η)
`=1
where ψ (`)(ξ, η) are the 2D Legendre polynomials:
(
)
√
√
√
√
5 2
5 2
(`)
ψ (ξ, η) ∈ 1, 3ξ, 3η,
(3ξ − 1), 3ξη,
(3η − 1), ...
2
2
where
M1
M0 ;
Multiscale Solver Results: Shock Tube Problem
x 1 ,x 1 × v 1 ,v 1
i− 2 i+ 2
j− 2 j+ 2
v f dv
−∞
u = R∞
−∞ f dv
Selected References
∆v
=
R∞
Solve with frozen q :
 


ρu
ρ
 ρu  + 
 =0
ρu2 + p
E ,t
u(E + p) + 12 q ,x
2
ρ
M=p
2πpn+1
Figure 3: Fluid solver results for the shock tube problem. From left: ρ = M0; u =
M (M +1)
2
X
Get heat fluxZ from kinetic distribution:
∞
q=
(v − u)3 f dv
3
n+1
d
a

∆t time-step, start at tn:  (ρu)n 
En
I
`=1
c
ρn
−∞
∆x
Dividing our domain into a mesh of
equally sized cells with x ∈ [a, b] and
v ∈ [c, d]:

Figure 2: Solution to shock tube problem, showing three different types of waves.
(`)
and Fi (t) is the L2 projection:
Z 1
1
∆x
(`)
(`)
Fi (t) =
φ (ξ) f t, xi +
ξ dξ
2 −1
2
(2-Dimensional)
Multiscale Solver
I
x=0
x→
Figure 5: Before: u with Gibb’s phenomenon. After: u with slope limiter applied.
M
X
(`)
=
Fi (t) φ(`)(ξ)
where φ(`)(ξ) are the 1D Legendre polynomials:
)
(
√
√
5 2
(`)
(3ξ − 1), ...
where
φ (ξ) ∈ 1, 3ξ,
2
I
For compressible Euler equations, use characteristic variables
I
t=0
i−1
which are related to f as follows:
Z
Z
Z
1
ρ = f dv, u =
vf dv, p = (v − u)2 f dv,
ρ
I
I
F (`)(t) ψ (`)(ξ, η)
Solving 1D advection equation along each line using Semi-Lagrangian time-stepping:
and
I
Limiting slope:
1
replace
(1)
(1)
(1)
(1)
(2)
(2) 1
−−−−→ Qi
minmod Qi , √ Qi − Qi−1 , √ Qi+1 − Qi
3
3
where
sign(a) min(|a| , |b| , |c|)
if a,b,c have the same sign
minmod(a, b, c) =
0
otherwise

ρu
ρ
 ρu  +  ρu2 + p  = 0
E ,t
u(E + p) ,x
ρ := density,
x→
I
`=1
`=1
+
I The fluid limit, ε → 0 , of this kinetic model is the compressible Euler system:

Tij
Projecting 2D mesh onto a series of 1D lines:


M (M +1)
Z 1
2
X


1
(k)
(k)1D
(`)
(`)
∗
(k)

G =F
=
F (t) ψ (ξ, η )
φ (ξ) 
dξ

2 −1
I
I
≈
q↑
x→
M (M +1)
2
X
=⇒
q↑
(`)n+1
(`)n
(`)n
(`)n − ∆t
Fij
= Mij + Fij − Mij
e ε
ε → 0+
I
Our basic method produces a solution with overshoots and undershoots in slope
(Gibb’s phenomenon), which may be reduced by a slope limiter:
1 ∆t time-step on f = 1 (M − f )
,t
ε
2
∆t
−
n+1
f
(x, v) = M(x, v) + (f (x, v) − M(x, v))e ε
ε → 0+
I
I
1
f,t = (M − f )
ε
Exact Solution to
f,t + vf,x = 1ε (M − f )
Slope Limiter
2 (x − x ),
ξ = ∆x
i
2 (v − v )
η = ∆v
i
(`)
and Fij (t) is the L2 projection:
Z 1
1
∆x
∆v
(`)
(`)
Fij (t) =
ψ (ξ, η) f t, xi +
ξ, vj +
η dξ dη
4 −1
2
2
Joint work with: Dr. James Rossmanith and Pierson Guthrey
Shock tube initial condition with discontinuity at
x = 0:
   
ρ`
1
left state:  u`  =  0 
p`
1




ρr
0.125
right state:  ur  =  0 
pr
0.1
Figure 4: Left: Strongly collisional solution. Right: Weakly collisional solution.
5Supported by NSF Grant DMS-1457443
1
Lewis & Clark College (OR), 2Augustana College (SD), 3Marist College (NY), 4Iowa State University (IA)