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 Wenjun Sun, Kun Xu) Song Jiang Nonlinear stability for MHD flows

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 that goes 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 leading-order term is said to be the asymptotic solution of the original PDE, and the simpler PDE the asymptotic limit. • In many instances, the scale lengths associated with solutions of 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 • Outline: 1. Governing equations 2. An AP scheme for a linear equation 3. Asymptotic analysis, AP property 4. AP scheme for the system 5. Numerical experiments 6. conclusions 7. Future studies Song Jiang Nonlinear stability for MHD flows

1. Governing equations Grey radiative transfer equations: ǫ 2 ∂ I Ω · ∇ I = σ ( 1 4 π acT 4 − I ) , ∂ t + ǫ� (1) c ∂ T � � Ω − acT 4 � Id � ǫ 2 C V ∂ t = σ . (2) r , � I ( � Ω , t ) : Radiation intensity, T ( � r , t ) : Material temperature, σ ( � r , T ) : Opacity, a : Radiation constant, c : Speed of light, ǫ > 0: Knudsen number, C V ( � r , t ) : Specific heat, r : Spatial variable, � � 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 ) = 4 π ac ( T ( 0 ) ) 4 , and a diffusion equation 1 ∂ t T ( 0 ) + a ∂ ∂ ∂ t ( T ( 0 ) ) 4 = ∇ · ac 3 σ ∇ ( T ( 0 ) ) 4 . C V (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

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, · · · ’93-’13 • 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) 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 Construction strategy of an AP scheme for (1), (2): i. Discretize (1) explicitly by a UGKS (discrete ordinate method like) and show it to be AP; ii. Take moments to (1) and combine it with (2) to give a macro diffu- sion system, and solve this system implicitly; iii. Resolve (1) by inserting the renewed values obtained by ii; iv. Resolve (2) by inserting the renewed values obtained by i. Song Jiang Nonlinear stability for MHD flows

2. AP scheme for the system (1)–(2) 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: � 1 ǫ c ∂ t f + µ∂ x f + ξ∂ y f = σ � 2 π φ − f − ǫα f + ǫ G , (4) ǫ 1 − ζ 2 cos θ , � where φ ( t , x , y ) , G ( x , y ) : given functions, µ = 1 − ζ 2 sin θ , � ξ = ζ ∈ [ 0 , 1 ] : cosine value of the angle between the propagation direction � Ω and z -axis, θ ∈ [ 0 , 2 π ) : 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 ( µ m , ξ m ) , m = 1 , · · · , M ( = N ( N + 2 ) / 2) = ⇒ direction discrete equation: ǫ c ∂ t f m + µ m ∂ x f m + ξ m ∂ y f m = σ ǫ ( 1 2 π φ − f m ) − ǫα f m + ǫ G . (5) i , j , m : cell average of f m at t n in cell ( i , j ) := { ( x , y ); x i − 1 / 2 < Denote f n x < x i + 1 / 2 , y j − 1 / 2 < y < y j + 1 / 2 } , integrate (5) = ⇒ FV discretization of (5) reads as i , j , m + ∆ t ∆ x ( F i − 1 / 2 , j , m − F i + 1 / 2 , j , m ) + ∆ t f n + 1 i , j , m = f n ∆ y ( G i , j − 1 / 2 , m − G i , j + 1 / 2 , m ) � σ � φ i , j − ˜ f i , j , m ) − α ˜ f i , j , m + ˜ ǫ 2 ( ˜ + c ∆ t G i , j , (6) F i − 1 / 2 , j , m , G i , j − 1 / 2 , m : numerical fluxes in the x -, y -directions, and Song Jiang Nonlinear stability for MHD flows

� t n + 1 1 f m ( t , x i − 1 / 2 , y j , µ m , ξ m ) dt , F i − 1 / 2 , j , m = ∆ t t n � t n + 1 1 f m ( t , x i , y j − 1 / 2 , µ m , ξ m ) dt , G i , j − 1 / 2 , m = (7) ∆ t t n φ i , j , ˜ ˜ f i , j , m , ˜ G i , j denote averages of φ, f m , G in ( t n , t n + 1 ) × cell ( i , j ) . we have to give the expressions in (7) explicitly. First, for ˜ f i , j , m we take (implicit in time): ˜ f i , j , m ≈ f n + 1 i , j , m . To evaluate F i − 1 / 2 , j , m , we solve (5) in the x -direction at the cell boundary x = x i − 1 / 2 , y = y j : ǫ c ∂ t f m + µ m ∂ x f m = σ ǫ ( 1 2 π φ − f m ) − ǫα f m + ǫ G , f m ( x , y j , t ) | t = t n = f m 0 ( x , y j , t n ) (8) Song Jiang Nonlinear stability for MHD flows

to have the explicit solution 0 ( x i − 1 / 2 − c µ m f m ( t , x i − 1 / 2 , y j , µ m , ξ m ) = e − ν i − 1 / 2 , j ( t − t n ) f m ( t − t n )) ǫ � t t n e − ν i − 1 / 2 , j ( t − s ) c σ i − 1 / 2 , j φ ( s , x i − 1 / 2 − c µ m + ( t − s )) ds 2 πǫ 2 ǫ + c ( 1 − e − ν i − 1 / 2 , j ( t − t n ) ) G , (9) ν i − 1 / 2 , j where ν = c ( σ ǫ 2 + α ) , ν i − 1 / 2 , j : value of ν at the corresponding cell boundary. Put (9) into (7) ⇒ numerical flux F i − 1 / 2 , j , m in x , provided the initial data f 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 G i , j − 1 / 2 , m in y can be constructed in the same manner) Song Jiang Nonlinear stability for MHD flows

• Approximate f m 0 ( x , y j , t n ) by piecewise constants: � f n i − 1 , j , m , if x < x i − 1 / 2 , f m 0 ( x , y j , t n ) = (10) f n i , j , m , if x > x i − 1 / 2 . • Evaluate φ ( x , y , t ) by piecewise polynomials:  φ n + 1 i − 1 / 2 , j + δ x φ n + 1 , L i − 1 / 2 , j ( x − x i − 1 / 2 ) , if x < x i − 1 / 2 ,  φ ( x , y j , t ) = (11) φ n + 1 i − 1 / 2 , j + δ x φ n + 1 , R i − 1 / 2 , j ( x − x i − 1 / 2 ) , if x > x i − 1 / 2 .  Here φ n + 1 i − 1 / 2 , j to be given later, φ n + 1 i − 1 / 2 , j − φ n + 1 φ n + 1 − φ n + 1 i − 1 , j i , j i − 1 / 2 , j δ x φ n + 1 , L , δ x φ n + 1 , R i − 1 / 2 , j = i − 1 / 2 , j = . ∆ x / 2 ∆ x / 2 ∼ boundary flux in y can be constructed in the same way. Song Jiang Nonlinear stability for MHD flows

3. Asymptotic analysis, AP property • The above constructed scheme is AP By (9)-(11), flux F i − 1 / 2 , j , m can be decomposed into F i − 1 / 2 , j , m = A i − 1 / 2 , j µ m ( f n i − 1 , j , m 1 µ m > 0 + f n i , j , m 1 µ m < 0 ) � m δ x ρ n + 1 , L + C i − 1 / 2 , j µ m ρ n + 1 µ 2 i − 1 / 2 , j + D i − 1 / 2 , j i − 1 / 2 , j 1 µ m > 0 (12) � m δ x ρ n + 1 , R + µ 2 i − 1 / 2 , j 1 µ m < 0 + E i − 1 / 2 , j µ m G , where ν = c ( α + σ/ǫ 2 ) , 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 � t n + 1 c µ � ( Diff ) n + 1 i − 1 / 2 , j := f ( t , x i − 1 / 2 , y j , µ, ξ ) dtd µ 2 πǫ ∆ t t n M = 1 � ω m µ m f m ( t , x i − 1 / 2 , y j , µ m , ξ m ) 2 π m = 1 φ n + 1 − φ n + 1 1 i − 1 , j i , j − ǫ → 0 − − → , 3 σ i − 1 / 2 , 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