An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations

Transcription

1 An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations Institute of Applied Physics and Computational Mathematics, Beijing NUS, Singapore, March 2-6, 2015 (joint work with Wenjun Sun, Kun Xu)

2 Introduction An asymptotic limit associated with a PDE is a limit in which certain terms in the PDE are made small relative to other terms. This ordering in size is achieved via a scaling parameter (say ɛ) that goes to zero in the asymptotic limit. In many instances, the scale lengths associated with solutions of the limiting 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 completely impractical mesh resolution requirement ( x O(ɛ), ɛ 0).

3 Aim of this talk: transfer system To present an AP scheme for the grey radiative Outline: 1. Governing equations 2. An AP scheme for the system 3. Asymptotic analysis, AP property 4. Numerical experiments 5. conclusions 6. Future studies

4 1. Governing equations Grey radiative transfer equations: ɛ 2 c I t + ɛ Ω I = σ( 1 4π act 4 I), (1) ɛ 2 C V 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, ɛ > 0: Knudsen number, C V ( r, t): Specific heat, r: Spatial variable, Ω: Angular variable, t: Time variable. σi: absorption term; σ 4π act 4 : emission term.

5 Backgrounds

6 Numerical challenges High dimensional problem time: t, spatial: r, direction : Ω. For multi-material problems (multi-scale) optically thick material: small ɛ optically thin material: large ɛ To resolve physical scales, normally x O(ɛ/σ) huge computing costs for small ɛ For small ɛ, the radiation transfer behaves like diffusion how to construct a scheme which can capture the diffusion behavior for small ɛ? ( asymptotic preserving)

7 Formally, as ɛ 0 (optically thick), (1)-(2) behaves like a diffusion equation. Expand I and T in ɛ and insert the expansions into (1)-(2) the leading-order term satisfies (formally): I (0) = 1 4π ac(t (0) ) 4, and a diffusion equation C V 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 ɛ/σ small.

8 Related results: AP schemes were introduced first by Larsen, Morel and Miller 87 for steady neutron transport problems Further developments for different non-steady problems, based on a decomposition of the distribution function between an equilibrium part and its derivation, by Klar, Jin, Pareschi, Toscani, A different approach by Xu & Huang 10 for rarefied gas dynamics based on a unified gas kinetic scheme (UGKS), further development by Mieussens 13 for a linear transport model. (scalar equations are dealt with only, not coupled with other eqs.) We want to use the idea of gas kinetic schemes to construct an AP scheme for the system (1)-(2).

9 2. AP scheme for the system (1) (2) 2.1. Angular discretization In 2D Cartesian coordinates, we can rewrite (1), (2) as ɛ c ti + µ x I + ξ y I = σ ( 1 ) ɛ 2π φ I, ɛ 2 T C V = σ( Id ) Ω φ, φ = act 4. t where µ = 1 ζ 2 cos θ, ξ = 1 ζ 2 sin θ, 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. (4)

10 2.1. Angular discretization We use the usual discrete ordinate method to discretize (4) in directions. Write the propagation direction (µ, ξ) as some discrete directions (µ m, ξ m ), m = 1,, M (= N(N + 2)/2), together with the corresponding integration weights ω m = angular discretized equations: ɛ c ti m + µ m x I m + ξ m y I m = σ ɛ ( 1 2π φ I m), ɛ 2 C V T t ( M = σ m=1 ) I m ω m φ, φ = act 4. (5)

11 2.2. Time and spatial discretization Denote I n,m, T n : cell average values of I m, T at t n in cell (i, j) := {(x, y); x i 1/2 < x < x i+1/2, y j 1/2 < y < y j+1/2 }, and integrate (5) FV discretization of (5) reads as I n+1,m = I n,m + t x (F i 1/2,j,m F i+1/2,j,m ) + t y (H 1/2,m H +1/2,m ) + c t σ ɛ 2 ( 1 2π φ Ĩ,m ), (6) C v T n+1 = C v T n + t σ ɛ ( M Ĩ 2,m ω m φ ), m=1 where F i 1/2,j,m, H 1/2,m : numerical fluxes in the x-, y-directions, φ, Ĩ,m : average of φ, I m in (t n, t n+1 ) cell(i, j); and

12 F i±1/2,j,m = cµ t n+1 m ɛ t H ±1/2,m = cξ m ɛ t t n t n+1 I m (t, x i±1/2, y j, µ m, ξ m )dt, t n I m (t, x i, y j±1/2, µ m, ξ m )dt, (7) and we take here approximation (implicitly in time): Ĩ,m I n+1,m, φ φ n+1. We have to construct the numerical fluxes in (7).

13 2.3. Construction of the numerical fluxes To evaluate F i 1/2,j,m, we use the idea of gas kinetic schemes, i.e., solve (5) in the x-direction at the cell boundary x = x i 1/2, y = y j : to have the explicit solution ɛ c ti m + µ m x I m = σ ɛ ( 1 2π φ I m), I m (x, y j, t) t=t n = I m,0 (x, y j, t n ) (8) I m (t, x i 1/2, y j, µ m, ξ m ) = e ν i 1/2,j (t t n) I m,0 (x i 1/2 cµ m ɛ (t t n )) + t t n e ν i 1/2,j (t s) cσ i 1/2,j 2πɛ 2 φ(s, x i 1/2 cµ m (t s))ds, (9) ɛ where ν = cσ/ɛ 2, ν i 1/2,j : value of ν at the corresponding cell boundary. (I m,0, φ to be determined)

14 Put (9) into (7) numerical flux F i 1/2,j,m in x, provided the initial data I m,0 (x, y j, t n ) in (8), φ(x, y, t) for t (t n, t n+1 ) and (x, y) around (x i 1/2, y j ) in (9) are known. (Flux H 1/2,m in y can be constructed in the same manner) Approximate I m,0 (x, y j, t n ) explicitly by piecewise linear polynomials and MUSCL limiter for slopes, using known I,m n

15 Evaluate φ(x, y, t) implicitly by piecewise polynomials: φ(x, y j, t) = φ n+1 i 1/2,j + δ tφ n+1 i 1/2,j (t t n+1 ) δ x φ n+1,l i 1/2,j + (x x i 1/2), if x < x i 1/2, δ x φ n+1,r i 1/2,j (x x i 1/2), if x > x i 1/2. (10) Here δ t φ n+1 i 1/2,j = (φ n+1 i 1/2,j φn i 1/2,j )/ t, and δ x φ n+1,l i 1/2,j = φn+1 i 1/2,j φn+1 i 1,j x/2, δ x φ n+1,r i 1/2,j = φn+1 φ n+1 x/2 i 1/2,j The numerical flux F i 1/2,j,m is obtained, provided the implicit terms φ n+1 i 1/2,j, φn+1 i 1,j, φn+1 are evaluated..

16 Remark. By (9) (10), flux F i 1/2,j,m can be decomposed into F i 1/2,j,m = A i 1/2,j µ m (I i 1/2,j,m 1 µ m>0 + I + i 1/2,j,m 1 µ m<0) +C i 1/2,j µ m φ n+1 i 1/2,j +D i 1/2,j ( µ 2 m δ xφ n+1,l i 1/2,j 1 µ m>0 + µ 2 m δ xφ n+1,r i 1/2,j 1 µ m<0) (11) +B i 1/2,j (µ 2 m δ xi n i 1,j,m 1 µ m>0 + µ 2 m δ xi n,m 1 µ m<0) +E i 1/2,j µ m δ t φ n+1 i 1/2,j, (involved the values of I at t n only) where I i 1/2,j,m, I + i 1/2,j,m denote the boundary values: I i 1/2,j,m = I n i 1,j,m + x 2 δ xi n i 1,j,m, I + i 1/2,j,m = I n,m x 2 δ xi n,m, and ν = cσ/ɛ 2 ; the coefficients A i 1/2,j = are explicit functions of t, ɛ, σ, ν.

17 2.4. Evaluation of φ n+1 in the flux F i 1/2,j,m To evaluate φ n+1, we take the moment of (1). Denote φ = act 4, ρ = IdΩ as before, take angular moment of (1) (1), (2) can be rewritten in the macro-form: ɛ 2 ρ c t + ɛ < ΩI >= σ(φ ρ), < ΩI >:= ΩId Ω, ɛ 2 φ t = βσ(ρ φ), β = 4acT 3 C V. We discretize (12) implicitly as follows. ρ n+1 = ρ n + t x (Φn+1 i 1/2,j Φn+1 i+1/2,j ) (12) + t y (Ψn+1 1/2 Ψn+1 +1/2 ) + cσn+1 t (φ n+1 ɛ 2 ρ n+1 ), (13) φ n+1 = φ n + (βσ)n+1 t (ρ n+1 ɛ 2 φ n+1 ),

18 where Φ n+1 i±1/2,j = Ψ n+1 ±1/2 = c ɛ t c ɛ t t n + t t n t n + t t n < Ω x I > (x i±1/2, y j, t)dt, < Ω y I > (x i, y j±1/2, t)dt, which can be explicitly evaluated using (9) and (11), e.g., Φ n+1 i 1/2,j = M ω m F i 1/2,j,m = A i 1/2,j m=1 +I n,m 1 µ m<0 ) + 2πD i 1/2,j 3 M m=1 ( φn+1 φ n+1 i 1,j x ω m µ m (I n i 1,j,m 1 µ m>0 M +B i 1/2,j ω m µ 2 m (δ xi n i 1,j,m 1 µ m>0 + δ x I n,m 1 µ m<0), m=1 ) where A i 1/2,j, are same as before ( only involved the values of I at t n ) (σ i 1/2,j = 2σ σ i 1,j σ +σ i 1,j )

19 (13) is a nonlinear system for φ n+1, ρ n+1 with parameters σ, β depending on T, and is solved by a two-level iteration method, outer iteration: a nonlinear iteration with fixed σ, β a linear algebraic system; inner iteration: Gauss-Sidel iteration to solve the linear system. After obtaining φ n+1, we take the cell boundary value φ n+1 i 1/2,j in (10) to be φ n+1 i 1/2,j = 1 2 (φn+1 i 1,j + φn+1 ). (14) With (14), the numerical fluxes (10) are determined. The construction of our AP UGKS scheme is complete

20 LOOP of the AP-UGKS: Given I n,m and T n, we have ρn and φn. Find I n+1 and T n+1. 1) Solve the system (13) to obtain φ n+1, ρ n+1 ; 2) Using φ n+1 to solve the system (6) for I to get I n+1,m, T n+1 ; 3) (Corrector step) Using I n+1,m to solve (12) 2 (i.e. ɛ 2 φ t to get a new φ n+1 ( explicitly given by = βσ(ρ φ)) φ n+1 = φn + M t(βσ)n+1 m=1 ω mi n+1,m 1 + t(βσ) n+1 ). And then, to give the corrected temperature by T n+1 = ( φ n+1 /(ac)) 1/4. 4) Goto 1) for the next computational step.

21 3. Asymptotic analysis, AP property The above constructed scheme is AP Recalling (11), F i 1/2,j,m can be decomposed into F i 1/2,j,m = A i 1/2,j + + E i 1/2,j, where the coefficients satisfy A, B 0, D c/(2πσ) as ɛ 0. Thus, the corresponding macroscopic diffusion flux (Diff) n+1 i 1/2,j : (Diff) n+1 i 1/2,j := = 1 2π M m=1 cµ 2πɛ t t n+1 t n I(t, x i 1/2, y j, µ, ξ)dtdµdξ ω m µ m I m (t, x i 1, y j, µ m, ξ m ) c φ n+1 φ n+1 2 3σ n+1 x i 1 2,j i 1,j as ɛ 0, which gives a numerical flux of the limiting diffusion equation for φ 5 points scheme in 2D an AP scheme.

22 5. Numerical experiments Example 1. A test for the linear kinetic equation: ɛ c ti + µ x I = σ ɛ ( 1 2π φ I ), where we take φ = Idµ, Domain: x [0, 1], σ = 1, ɛ = 10 8 ; boundary conditions: I L (0, µ) = 1 (µ > 0 and I R (1, µ) = 0 (µ < 0); x = , >> ɛ. This problem corresponds to the equilibrium diffusion approximation.

23 Computed solution at t = 0.01, 0.05, 0.15, 2.0: 0.8 circle: UGKS method with 200 points diamond: UGKS method with 50 points square : diffusion limit solution with 200 points 0.6 E x Computed solution agrees well with that of the diffusion limiting eq. AP property

24 Example 2. (Marshak wave-2b) σ = 100 T 3 cm 2 /g, T = 10 6 Kev, ɛ = 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 >> ɛ/σ. Left boundary: constant incident radiation intensity with a Planckian distribution at 1Kev; Right boundary: outflow

25 Computed material temperature at t = 15, 30, 45, 60, 74ns 0.8 square : t = 15ns; Delta : t= 30ns; Diamond : t= 45ns; Circle: t=60ns; Gradient: t=74ns. 0.6 T x

26 Computed material temperature for both grey radiation transfer system and diffusion limiting equation at 74ns T Circle : Diffusion limit solution with 200 points; Diamond : UGKS solution with 200 points x Results computed by AP UGKS agree very well with numerical solution of the diffusion limiting eq. AP property

27 Example 3. (Marshak wave-2a) the same as Example 2 except σ = 10 T 3 cm 2 /g. This case violates the equilibrium diffusion approximation. Computed radiative temperature at t = 0.2, 0.4, 0.6, 0.8, 1.0ns: square : t = 0.2ns; Delta : t= 0.4ns; Diamond : t= 0.6ns; Circle: t=0.8ns; Gradient: t=1.0ns. T x

28 Material temperature distributions obtained by AP UGKS and the diffusion equation solution at t = 1.0ns: (clear difference, since not in the equilibrium diffusion approximation) 0.8 T Circle : Diffusion limit sloution with 200 points; Diamond: UGKS solution with 200 points. 0 x

29 Example 4. (Tophat Test [N.A. Gentile, 01]) Domain: [0, 7] [ 2, 2], ɛ = 1; Five probes: placed at (0.25, 0), to monitor the change of the temperature in the thin opacity material.

30 Computed material temperature at 8ns: cells cells Interface between optically thick and thin materials is captured sharply, compared with [Gentile 01 (JCP)]

31 Computed radiation temperature at 8ns: cells cells Interface between optically thick and thin materials is captured sharply.

32 Computed material temperature at 94ns: cells cells

33 Time evolution of the material, radiation temperatures at 5 probes: 0.4 Temperature(KEV) time(10ns) At the 5th probe, the temperature cools off slightly before being heated up by radiation, which is consistent with [Gentile]. But, the temperature curve here has a rapid growth initially, a slow increment, then a growth again. This is different from [Gentile], deserving further investigation.

34 6. Conclusions A unified gas scheme is constructed for the grey radiative transfer system. The scheme has asymptotic preserving property and works well for both optically thin and optically thick regimes. We believe that it should work for more general problems, say, the multi-group radiation transfer equations 7. Future studies Extension to the multi-group radiation transfer equations Extension to the multi-material problems, quadrilateral meshes Extension to radiation hydrodynamics Question: An AP scheme converges to the desired diffusion solver?

35 THANK YOU!