An asymptotic preserving unified gas kinetic scheme for the grey - - PowerPoint PPT Presentation
An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations Song Jiang Institute of Applied Physics and Computational Mathematics, Beijing China-Gemany Workshop, May 23-26, 2014, Beijing (joint work with
Song Jiang
Institute of Applied Physics and Computational Mathematics, Beijing
China-Gemany Workshop, May 23-26, 2014, Beijing (joint work with Wenjun Sun, Kun Xu)
Song Jiang Nonlinear stability for MHD flows
terms in the PDE are made “small” relative to other terms.
to zero in the asymptotic limit. For instance, denoting the scaling parameter by ǫ, a power-series solution in ǫ is assumed for the scaled PDE. Inserting the expansion into the PDE and equating coefficients of each power of ǫ lead to a heirarchical set of equations. One generally finds that the leading-order term satisfies a “sim- pler” PDE than the original one.
Song Jiang Nonlinear stability for MHD flows
the asymptotic equation are much larger than the smallest scale lengths associated with solutions of the original PDE. When this is the case, asymptotic-preserving (AP) discretization schemes are necessary for near-asymptotic problems to avoid com- pletely impractical mesh resolution requirements (∆ x ∼ O(ǫ), ǫ→0).
Song Jiang Nonlinear stability for MHD flows
Aim of this talk: To present an AP scheme for the grey radiative transfer system
Song Jiang Nonlinear stability for MHD flows
ǫ2 c ∂I ∂t + ǫ Ω · ∇I = σ( 1 4π acT 4 − I), (1) ǫ2CV ∂T ∂t = σ Id Ω − acT 4 . (2) I( r, Ω, t): Radiation intensity, T( r, t): Material temperature, σ( r, T): Opacity, a: Radiation constant, c: Speed of light, ǫ > 0: Knudsen number, CV ( r, t): Specific heat,
Ω: Angular variable, t: Time variable. Formally, as ǫ → 0 (opatically thick), (1)-(2) behaves like a diffusion equation.
Song Jiang Nonlinear stability for MHD flows
Generally, a computational domain contains both small ǫ and large ǫ, leading to numerial diffuculties, since ∆x ∼ O(ǫ/σ) to resolve physical scale huge computing costs for small ǫ/σ. Expand I and T in ǫ and insert the expansions into (1)-(2) ⇒ the leading-order term satisfies (formally): I(0) =
1 4πac(T (0))4, and a diffusion equation
CV ∂ ∂t T (0) + a ∂ ∂t (T (0))4 = ∇ · ac 3σ ∇(T (0))4. (3) An asymptotic preserving (AP) scheme for (1)-(2) is a numerical scheme that discretizes (1)-(2) in such a way that it leads to a correct discretization of the diffusion limit (3) when ǫ/σ small.
Song Jiang Nonlinear stability for MHD flows
for steady neutron transport problems
a decomposition of the distribution function between an equilibrium part and its derivation, by Klar, Jin, Pareschi, Toscani, · · · ’93-’13
based on a unified gas kinetic scheme (UGKS), further development by Mieussens ’13 for a linear transport model. (scalar equations are dealt with only) We want to use the idea of gas kinetic schemes to construct an AP scheme for the system (1)-(2).
Song Jiang Nonlinear stability for MHD flows
Construction strategy of an AP scheme for (1), (2):
and show it to be AP;
sion system, and solve this system implicitly;
Song Jiang Nonlinear stability for MHD flows
2.1. A UGKS for the radiative transfer equation (1) We shall solve (1) and (2) respectively. To solve (1), we taken T to be known. W.l.g., consider a general 2D linear kinetic equation: ǫ c ∂tf + µ∂xf + ξ∂yf = σ ǫ 1 2π φ − f
(4) where φ(t, x, y), G(x, y): given functions, µ =
ξ =
ζ ∈ [0, 1]: cosine value of the angle between the propagation direction Ω and z-axis, θ ∈ [0, 2π): projection vector of Ω onto the xy-plane and the x-axis.
Song Jiang Nonlinear stability for MHD flows
To discretize (4), as the usual discrete ordinate method, write the propagation direction (µ, ξ) as some discrete directions (µm, ξm), m = 1, · · · , M (= N(N + 2)/2) = ⇒ direction discrete equation: ǫ c ∂tf m + µm∂xf m + ξm∂yf m = σ ǫ ( 1 2π φ − f m) − ǫαf m + ǫG. (5) Denote f n
i,j,m: cell average of f m at tn in cell (i, j) := {(x, y); xi−1/2 <
x < xi+1/2, yj−1/2 < y < yj+1/2}, integrate (5) = ⇒ FV discretization of (5) reads as f n+1
i,j,m = f n i,j,m + ∆t
∆x (Fi−1/2,j,m − Fi+1/2,j,m) + ∆t ∆y (Gi,j−1/2,m − Gi,j+1/2,m) +c∆t σ ǫ2 ( ˜ φi,j − ˜ fi,j,m) − α˜ fi,j,m + ˜ Gi,j
(6) Fi−1/2,j,m, Gi,j−1/2,m: numerical fluxes in the x-, y-directions, and
Song Jiang Nonlinear stability for MHD flows
Fi−1/2,j,m = 1 ∆t tn+1
tn
f m(t, xi−1/2, yj, µm, ξm)dt, Gi,j−1/2,m = 1 ∆t tn+1
tn
f m(t, xi, yj−1/2, µm, ξm)dt, (7) ˜ φi,j, ˜ fi,j,m, ˜ Gi,j denote averages of φ, f m, G in (tn, tn+1) × cell(i, j). we have to give the expressions in (7) explicitly. First, for ˜ fi,j,m we take (implicit in time): ˜ fi,j,m ≈ f n+1
i,j,m.
To evaluate Fi−1/2,j,m, we solve (5) in the x-direction at the cell boundary x = xi−1/2, y = yj: ǫ c ∂tf m + µm∂xf m = σ ǫ ( 1 2π φ − f m) − ǫαf m + ǫG, f m(x, yj, t)|t=tn = f m
0 (x, yj, tn)
(8)
Song Jiang Nonlinear stability for MHD flows
to have the explicit solution f m(t, xi−1/2, yj, µm, ξm) = e−νi−1/2,j(t−tn)f m
0 (xi−1/2 − cµm
ǫ (t − tn)) + t
tn e−νi−1/2,j(t−s) cσi−1/2,j
2πǫ2 φ(s, xi−1/2 − cµm ǫ (t − s))ds +c(1 − e−νi−1/2,j(t−tn)) νi−1/2,j G, (9) where ν = c( σ
ǫ2 + α),
νi−1/2,j: value of ν at the corresponding cell boundary. Put (9) into (7) ⇒ numerical flux Fi−1/2,j,m in x, provided the initial data f m
0 (x, yj, tn) in (8), φ(x, y, t) for t ∈ (tn, tn+1) and (x, y) around
(xi−1/2, yj) in (9) are known. (• flux Gi,j−1/2,m in y can be constructed in the same manner)
Song Jiang Nonlinear stability for MHD flows
0 (x, yj, tn) by piecewise constants:
f m
0 (x, yj, tn) =
i−1,j,m,
if x < xi−1/2, f n
i,j,m,
if x > xi−1/2. (10)
φ(x, yj, t) = φn+1
i−1/2,j + δxφn+1,L i−1/2,j(x − xi−1/2), if x < xi−1/2,
φn+1
i−1/2,j + δxφn+1,R i−1/2,j(x − xi−1/2), if x > xi−1/2.
(11) Here φn+1
i−1/2,j to be given later,
δxφn+1,L
i−1/2,j =
φn+1
i−1/2,j − φn+1 i−1,j
∆x/2 , δxφn+1,R
i−1/2,j =
φn+1
i,j
− φn+1
i−1/2,j
∆x/2 . ∼ boundary flux in y can be constructed in the same way.
Song Jiang Nonlinear stability for MHD flows
By (9)-(11), flux Fi−1/2,j,m can be decomposed into Fi−1/2,j,m = Ai−1/2,j µm (f n
i−1,j,m1µm>0 + f n i,j,m1µm<0)
+Ci−1/2,j µm ρn+1
i−1/2,j + Di−1/2,j
m δxρn+1,L i−1/2,j1µm>0
(12) +µ2
m δxρn+1,R i−1/2,j1µm<0
where ν = c(α + σ/ǫ2), and A → 0, D → −1/σ as ǫ → 0.
Song Jiang Nonlinear stability for MHD flows
Thus, the corresponding macroscopic diffusion flux (Diff)n+1
i−1/2,j, de-
fined by (Diff)n+1
i−1/2,j :=
2πǫ∆t tn+1
tn
f(t, xi−1/2, yj, µ, ξ)dtdµ = 1 2π
M
ωmµmf m(t, xi−1/2, yj, µm, ξm) − − →
ǫ→0 −
1 3σi−1/2,j φn+1
i,j
− φn+1
i−1,j
∆x , which gives a numerical flux of the asymptotic limiting diffusion eq. for φ ∼ 3 resp. 5 points schemes in 1D resp. 2D ⇒ an AP scheme.
Song Jiang Nonlinear stability for MHD flows
We have constructed an AP scheme for (1), provided ˜ φi,j in (6) and φn+1
i−1/2,j in (11) to be determined.
Denote φ = acT 4, ρ =
Ω as before, take angular moment of (1) ⇒ (1), (2) can be rewritten in the macro-form: ǫ2 c ∂ρ ∂t + ǫ∇· < ΩI >= σ(φ − ρ), < ΩI >:=
Ω, ǫ2 ∂φ ∂t = βσ(ρ − φ). (13) We discretize (13) implicitly as follows. ρn+1
i,j
= ρn
i,j + ∆t
∆x (Φi−1/2,j − Φi+1/2,j) + ∆t ∆y (Ψi,j−1/2 − Ψi,j+1/2) + σc∆t ǫ2 (φn+1
i,j
− ρn+1
i,j ),
(14) φn+1
i,j
= φn
i,j + βσ∆t
ǫ2 (ρn+1
i,j
− φn+1
i,j ),
Song Jiang Nonlinear stability for MHD flows
where Φi±1/2,j = c ǫ∆t tn+∆t
tn
< ΩxI > (xi±1/2, yj, t)dt, Ψi,j±1/2 = c ǫ∆t tn+∆t
tn
< ΩyI > (xi, yj±1/2, t)dt, which can be explicitly evaluated using (9) and (12), e.g., Φi−1/2,j =
M
ωmFi−1/2,j,m = Ai−1/2,j
M
ωmµm
i−1,j,m1µm>0
+In
i,j,m1µm<0
3 ( φn+1
i,j
− φn+1
i−1,j
∆x ), where σi−1/2,j in Ai−1/2,j, Di−1/2,j is taken to be 2σi,jσi−1,j
σi,j+σi−1,j .
Song Jiang Nonlinear stability for MHD flows
(14) is a nonlinear system for φn+1
i,j , ρn+1 i,j
with parameters σ, β de- pending on T, and is solved by a two-level iteration method,
braic system; inner iteration: Gauss-Sidel iteration to solve the linear system. After obtaining φn+1
i,j , we take in (11) that
˜ φ = 1 2π φn+1
i,j ,
φn+1
i−1/2,j = 1
2(φn+1
i−1,j + φn+1 i,j ).
(15) With (15), the numerical fluxes (11) are determined. The construction of our AP UGKS scheme is complete
Nonlinear stability for MHD flows
Example 1. for the linear transport eq. (4), i.e., ǫ c ∂tf + µ∂xf = σ ǫ 1 2π φ − f
where we take φ =
ǫ = 10−8; boundary conditions: fL(0, µ) = 1 (µ > 0 and fR(1, µ) = 0 (µ < 0); ∆x = 5 × 10−3, 2.5 × 10−3 >> ǫ. This problem corresponds to the equilibrium diffusion approxima- tion.
Song Jiang Nonlinear stability for MHD flows
Computed solution at t = 0.01, 0.05, 0.15, 2.0:
x E
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
circle: UGKS method with 200 points diamond: UGKS method with 50 points square : diffusion limit solution with 200 points
Computed solution agrees well with that of the diffusion limiting eq. ∼ AP property
Song Jiang Nonlinear stability for MHD flows
Example 2. (Marshak wave-2B) σ =
100 T 3 cm2/g, T = 10−6Kev, ǫ = 1, specific heat=0.1GJ/g/Kev.
This corresponds to the equilibrium diffusion approximation. ∆x = 0.005cm (200 cells), ∆y = 0.01cm (1 cell), ∆x, ∆y >> ǫ/σ. Left boundary: constant incident radiation intensity with a Planckian distribution at 1Kev; Right boundary: outflow
Song Jiang Nonlinear stability for MHD flows
Computed material temperature at t = 15, 30, 45, 60, 74ns
x T
0.2 0.4 0.2 0.4 0.6 0.8
square : t = 15ns; Delta : t= 30ns; Diamond : t= 45ns; Circle: t=60ns; Gradient: t=74ns.
Song Jiang Nonlinear stability for MHD flows
Computed material temperature for both grey radiation transfer sys- tem and diffusion limiting equation at 74ns
V1 V2
0.2 0.4 0.2 0.4 0.6 0.8
Circle : Diffusion limit solution with 200 points; Diamond : UGKS solution with 200 points.
Results computed by AP UGKS agree very well with numerical solu- tion of the diffusion limiting eq. ∼ AP property
Song Jiang Nonlinear stability for MHD flows
Example 3. (Marshak wave-2A) the same as Example 2 except σ =
10 T 3 cm2/g. This case violates the equilibrium diffusion approximation.
Computed radiative temperature at t = 0.2, 0.4, 0.6, 0.8, 1.0ns:
x T
0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8
square : t = 0.2ns; Delta : t= 0.4ns; Diamond : t= 0.6ns; Circle: t=0.8ns; Gradient: t=1.0ns.
Song Jiang Nonlinear stability for MHD flows
Material temperature distributions obtained by AP UGKS and the dif- fusion equation solution at t = 1.0ns: (clear difference, since not in the equilibrium diffusion approximation)
V1 V2
0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8
Circle : Diffusion limit sloution with 200 points; Diamond: UGKS solution with 200 points.
Song Jiang Nonlinear stability for MHD flows
Example 4. (Tophat Test [N.A. Gentile,’01]) Domain: [0, 7] × [−2, 2], ǫ = 1; Five probes: placed at (0.25, 0), · · · to monitor the change of the temperature in the thin opacity material.
Song Jiang Nonlinear stability for MHD flows
Computed radiation temperature at 8ns and 94ns:
128 × 64 cells 256 × 128 cells
x y
1 2 3 4 5 6
1 2 3 4
x y
1 2 3 4 5 6
1 2 3 4
x y
1 2 3 4 5 6
1 2 3 4
V1 V2
1 2 3 4 5 6
1 2 3 4
Interface between the opacity thick and thin materials is captured sharply, comparing to [Gentile]
Song Jiang Nonlinear stability for MHD flows
Time evolution of the material and radiation temperatures at 5 probes:
t Temperature
1 2 3 4 5 6 7 8 9 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Diamond : material temperature; Circle : radiation temperture.
At the 5th probe, the temperature cools off slightly before being heated up by the radiation wave, which is physically reasonable and agrees with [Gentile]
Song Jiang Nonlinear stability for MHD flows
system.
both optical thin and optical thick regimes.
multi-group radiation transfer equations
Song Jiang Nonlinear stability for MHD flows
Remark: Cares have to be taken in the construction of AP schemes. Recently, Wu& Guo’14 considered the steady problem: ǫΩ · ∇f ǫ + f ǫ − 1 2π
in G, (16) f ǫ(x0, Ω) = g(x0, Ω) for Ω · n < 0, x0 ∈ ∂G, (17) where G = {x ∈ R2, |x| < 1}, Ω ∈ S1. They proved that f ǫ − leading-order termL∞ ≡ f ǫ − f 0 − f bL∞ = O(1), where f 0 is the corresponding interior solution to the Laplace eq. (dif- fusion eq.), f b is the Knudsen layer solution (to a Milne problem) ∼ no diffusion limit due to boundary. The correct limit is f ǫ − f 0
ǫ − f b ǫ L∞ = O(ǫ) no AP scheme.
Song Jiang Nonlinear stability for MHD flows
Song Jiang Nonlinear stability for MHD flows