A M A Mod odifi fied ed S Step ep Ch Character eristic M Method od fo for Solving the S N Tr Transport Equation Dean Wang The Ohio State University Zeyun Wu Virginia Commonwealth University 2019 ANS Winter Meeting, Washington DC, USA November 17-21, 2019
Outline • Background and motivation • Positivity or robustness • Accuracy • Diffusion limit • Modified step characteristic method (mSC) • Numerical formulation • A proof on the diffusion limit of SC (Wang 2019, NSE) • Numerical results • Conclusion 2
Finite difference sweeping methods Linear methods Ø Step difference (SD) • 1 st -order upwind • Positivity preserving • Intermediate diffusion limit, ∆𝑦 = 𝜁 % ℎ , where 𝑚 = 1 Ø Diamond difference (DD) • 2 nd -order • Not positivity preserving • Thick diffusion limit in interior homogeneous regions, 𝑚 = 0 Ø Step characteristic (SC) • Weighted DD • 2 nd -order, but less accurate than DD for diffusive problems • Positivity preserving • Intermediate diffusion limit, 𝑚 = 0 Nonlinear methods Ø LF-WENO methods (Wang 2019) • High-order • Very robust, but not positivity preserving. Can be made positive! • Between thick and intermediate, 𝑚 = 1/𝑙 , where 𝑙 is the order of spatial accuracy 3
SC 58< -,/ 54< -,/ , - . / 𝜔 1,345/6 − 𝜔 1,385/6 + Σ ;,3 𝜔 1,385/6 + 𝜔 1,345/6 = 6 6 58< -@,/ = >,/ 54< -@,/ G / B 6 ∑ 1 @ A5 𝜔 1 @ ,38 C D + 𝜔 1 @ ,345/6 𝑥 1F + 6 6 6 where 54I JKL,/M//N- 54I JO//N- 6, - 6, - 𝛽 1,3 = 58I JKL,/M//N- − = L,/ . / = 58I JO//N- − P / 𝜐 3 = Σ ;,3 ℎ 3 , cell optical thickness ℎ 3 = mesh size of cell 𝑘 4
M-matrix and stability B 𝜈 1 1 − 𝛽 1,3 𝜔 1,385/6 + 1 + 𝛽 1,3 𝜔 1,345/6 = Σ U,3 𝜔 1 @ ,3 𝑥 1F + 𝑅 3 𝜔 1,345/6 − 𝜔 1,385/6 + Σ ;,3 V 2 , ℎ 3 2 2 2 1 @ A5 𝑘 = 1, … 𝑛, for 𝜈 1 > 0 In matrix form: 𝐁𝛀 = 𝐓 where A is lower diagonal matrix ⋮ ⋮ ⋱ ⋮ ⋮ 1 − 𝛽 1,3 1 + 𝛽 1,3 − 𝜈 1 𝜈 1 … + Σ ;,3 + Σ ;,3 … … 𝐁 = ℎ 3 2 ℎ 3 2 ⋮ ⋮ ⋮ ⋱ ⋮ 58< -,/ , - For A is a M-matrix, we need to have − . / + Σ ;,3 ≤ 0 , and therefore 6 SD 𝛽 1,3 = 1 : A is unconditionally M-matrix 6, - DD 𝛽 1,3 = 0 : ℎ 3 ≤ d L,/ , or 𝜐 3 ≤ 2𝜈 1 > 1 6, - 6, - 5 SC: ℎ 3 ≤ d L,/ 58< -,/ , or 𝜐 3 ≤ 58< -,/ = 𝜐 3 , C 58 fe//N-JC N- e/ and therefore A is unconditionally M-matrix 5
Diffusion limit of S N – a recap B 𝑒𝑦 𝜔 1 + Σ ; 𝜔 1 = Σ U 𝑒 𝜔 1F 𝑥 1F + 𝑅 𝜈 1 2 V 2 1FA5 𝜯 𝒖 → 𝜯 𝒖 𝜯 𝒃 → 𝜻𝜯 𝒃 , 𝑹 → 𝜻𝑹 Scaling 𝜻 , B 𝑒𝑦 𝜔 1 + 𝛵 ; 𝑒 𝜁 𝜔 1 = 1 𝛵 ; 𝜔 1F 𝑥 1F + 𝜁𝑅 𝜈 1 𝜁 − 𝜁𝛵 u V 2 2 1FA5 n We have 𝜔 1 = 6 + 𝑃 𝜁 , for 𝜁 → 0 Where 𝜚 satisfies the following diffusion equation − 𝑒 1 𝑒 𝑒𝑦 𝜚 + Σ s 𝜚 = 𝑅 𝑒𝑦 3Σ q 6
Diffusion limit of SC (Wang, 2019) 8 = L/ w y . / x , - . / , - w yJC ⁄ = 1 + 𝑓 8 = L/ 𝛽 1,3 = 1 + 𝑓 w 2𝜈 1 2𝜈 1 − . / , - w yJC − Σ ;3 8 = L/ ⁄ Σ ;3 ℎ 3 𝜁 %85 1 − 𝑓 8 = L/ w y . / ⁄ , - 𝜁 % ℎ 3 1 − 𝑓 w 𝜁 Proof (by contradiction). If 𝑚 > 1 , then 𝛽 1,3 ↓ 0 as 𝜁 ↓ 0 , and thus SC tends to DD, whereas DD has 𝑚 = 0 . • If 𝑚 < 1 , then 𝛽 1,3 ↑ 1 for 𝜈 ~ > 0 , and 𝛽 1,3 ↓ −1 for 𝜈 1 < 0 , as 𝜁 ↓ 0 . Thus, SC tends • to SD, but SD has 𝑚 = 1 . 54I J KL/ M/ /N- 6, - So we should have 𝑚 = 1 for SC, and then 𝛽 1,3 = 58I J KL/ M/ /N- − • . = L/ . / 7
Modified SC (mSC) € /N- € JKL,/M/ CJ•/ JO/ CJ•/ x N- 54I 6, - 54I 6, - € , 𝛽 1,3 = € /N- − € = N- − € = L,/ . / 58• / P / 58• / JKL,/M/ CJ•/ JO/ CJ•/ x 58I 58I where 𝛾 is a positive number larger than 1 (e.g., 𝛾 = 3 ) = >,/ 𝑑 3 ≡ = L ,3 Note: 𝑑 ↓ 0: mSC → SC 𝑑 ↑ 1: mSC → DD 8
Diffusion limit of mSC (Wang, 2019) 8 = L/ € w y . / 58• / x , - 𝛽 1,3 = 1 + 𝑓 w 2𝜈 1 − Σ ;3 8 = L/ € … w y . / 58• / x , - 𝜁 % ℎ 3 1 − 𝑑 1 − 𝑓 w 𝜁 3 . / , - w yJC 58• / € = 1 + 𝑓 8 = L/ ⁄ 2𝜈 1 € − Σ ;3 ℎ 3 𝜁 %85 1 − 𝑑 … . / , - w yJC 58• / ⁄ 1 − 𝑓 8 = L/ 3 Proof . 3 = 1 − 𝜁 6 = †/ 𝑑 = L/ . • 𝜸 term tends to zero, and thus 𝛽 ↓ 0 . As a result, the SC As 𝜁 ↓ 0 , the 𝜻 𝒎8𝟐 𝟐 − 𝒅 𝒌 • reverts to the DD scheme, and therefore it can attain the thick diffusion limit as DD does. 9
How about positivity? € 𝛽 1,3 = 1 + 𝑓 8P / 58• / , - x 2𝜈 1 − € … 1 − 𝑓 8P / 58• / x , - τ 3 1 − 𝑑 3 For A is a M-matrix, we need to have € 58• / 6, - 𝜐 3 ≤ 58< -,/ = 𝐷𝜐 3 , where 𝐷 = € O/ CJ•/ x N- 58 € O/ CJ•/ x N-JC f c=0 0.2 0.4 0.6 0.8 0.9 0.99 100 10 𝐷 1 0.01 0.1 1 10 100 τ 3 / 𝜈 1 0.1 0.01 10
Numerical results – accuracy Σ q = 5 cm 85 𝑅 = 1 cm 85 𝑀 = 1 cm ℎ = 0.1 cm BC: Vacuum 11
Numerical Results – robustness 12
Numerical Results – diffusion limit L = 1 , h = 0.1, 5 5 Σ q = Σ • = ” − 0.8ε , ” , 𝑅 = ε , 13
Conclusions • We proposed a modified step characteristic method, called mSC, which can improve the accuracy of the original SC scheme. • The idea is that we have introduced a scaling factor, 1 − 𝑑 … in the weighting 𝛽 term of SC. • The numerical results have demonstrated that the new mSC scheme can preserve great robustness of the original SC, and is much more accurate than SC and DD as well. • More importantly it can attain the thick diffusion limit, which is of significant computational interest for thick diffusive problems such as radiative transfer. 14
References • K. D. LATHROP, “Spatial Differencing of the Transport Equation: Positivity vs. Accuracy,” J. Comput. Phys. , 4 , (1969). • E. W. Larsen, J. E. Morel, and W. F. Miller Jr., “Asymptotic Solutions of Numerical Transport Problems in Optically Thick, Diffusive Regimes,” J. Comput. Phys. , 69 , 283 (1987). https://doi.org/10.1016/0021- 9991(87)90170-7 • D. Wang, "The Asymptotic Diffusion Limit of Numerical Schemes for the S N Transport Equation," Nucl. Sci. Eng. , 193 , 12, 1339 (2019). https://doi.org/10.1080/00295639.2019.1638660 • D. Wang, T. Byambaakhuu, "High Order Lax-Friedrichs WENO Fast Sweeping Methods for the S N Neutron Transport Equation," Nucl. Sci. Eng. , 193, 9 , 982 (2019). https://doi.org/10.1080/00295639.2019.1582316 15
Thank You! 16
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