On a Recent Theoretical Result on Diffusion Limits of Numerical Methods for the S N Transport Equation in Optically Thick Diffusive Regimes Dean Wang The Ohio State University ICTT-26 Sorbonne University, Paris, France September 23-27, 2019
Outline • Background and motivation • Larsen et at., 1987 • Asymptotic analysis • Our result (Wang, 2019) • Local truncation error analysis • Remarks 2
Diffusion limit of S N – a recap 0 𝑒𝑦 𝜔 " + Σ ( 𝜔 " = Σ * 𝑒 𝜔 - 𝑥 - + 𝑅 𝜈 " 2 , 2 -./ Σ ( → 4 5 Σ 7 → 𝜁Σ 7 , 𝑅 → 𝜁𝑅 Scaling 6 , 𝑶 𝒆𝒚 𝝎 𝒏 + 𝜯 𝒖 𝒆 𝜻 𝝎 𝒏 = 𝟐 𝜯 𝒖 𝝎 𝒐 𝒙 𝒐 + 𝜻𝑹 𝝂 𝒏 𝜻 − 𝜻𝜯 𝒃 , 𝟑 𝟑 𝒐.𝟐 9 We have 𝜔 " = : + 𝑃 𝜁 , for 𝜁 → 0 Where 𝜚 satisfies the following diffusion equation − 𝑒 1 𝑒 𝑒𝑦 𝜚 + Σ B 𝜚 = 𝑅 𝑒𝑦 3Σ @ 3
Larsen et al.’s result (1987) ∆𝑦 = 𝜁 S ℎ A mesh size to resolve variations in the solution: 0 thick The three diffusion regimes are defined by 𝑚 = V 1 intermediate ≥ 2 thin 4
In IX. DISCUSSION, Larsen et al. propose “However, other choices of 𝑚 are possible and may be of interest. In particular, if 𝑚 is chosen between 0 and 1, then one has an asymptotic limit “between” the thick and intermediate limits considered above… Such choices of 𝑚 lead to curves in Fig. 1 that lie between the intermediate and thick (dashed) lines, and that approach the origin ( Δ𝑦, 𝜁) = (0, 0) tangent to the vertical axis.” 5
Our result (Wang, 2019) Δ𝑦 = 𝜁 ⁄ / h ℎ where 𝑙 is the order of accuracy of spatial discretization , and 𝑙 ≥ 1 𝑙 = 7 𝑙 = 3 6
How to … … Lo Local al tr trunc uncatio tion n error analy analysis is Asymptotic S N : 0 𝑒𝑦 𝜔 " + Σ ( 𝑒 𝜁 𝜔 " = 1 Σ ( 𝜔 - 𝑥 - + 𝜁𝑅 𝜈 " 𝜁 − 𝜁Σ 7 , 2 2 -./ 3 rd -order upwind method: Σ (l 𝜈 " 2𝜔 ",lm/ + 3𝜔 ",l − 6𝜔 ",ln/ + 𝜔 ",ln: + 𝜁 𝜔 ",l 6∆𝑦 l 0 Σ (l 𝜔 -,l 𝑥 - + 𝜁𝑅 l = 1 𝜁 − 𝜁Σ 7l , 2 2 -./ 7
Taylor expansion at the center of cell 𝑘 gives 4 5r 4 5r 6u r p / 0 (1) 6 − 𝜁Σ 7l ∑ -./ 𝜈 " pq 𝜔 ",l + 6 𝜔 ",l + 𝑩 = 𝜔 -,l 𝑥 - + : : where v ∆𝑦 l w + 𝑃 w 𝑩 = 𝜈 " 𝜔 ",l ∆𝑦 l 12 Let ∆𝒚 𝒌 = 𝜻 𝒎 𝒊 𝒌 , we have v 𝜁 S ℎ l w + 𝑃 w 𝜁 S ℎ l 𝑩 = 𝜈 " 𝜔 ",l 12 In order for Eq. (1) to approach the diffusion solution, we need to have 𝑩 ≪ 𝜁𝑅 l 2 8
v 𝜁 S ℎ l w ≪ 𝜁𝑅 l 𝜈 " 𝜔 ",l 12 2 w ≈ 𝑃 1 . In addition ℎ l ≈ If the solution is sufficiently smooth, then 𝜔 ",l 𝑃 1 , we have 𝜁 vS ≪ 𝜁 We obtain the greatest lower bound (or infimum) of 𝑚 : inf 𝑚 = 1/3 Generalizing to the 𝒍– 𝒖𝒊 order upwind method, we have inf 𝑚 = 1/𝑙 where 𝑙 is the spatial order of discretization 9
Numerical results LF-WENO3 10
Numerical results – manufactured solution 𝜔 𝑦, 𝜈 h = 𝑦 v 1 − 𝑦 v † 𝑅 𝑦 = 2 3𝑦 : − 12𝑦 v + 15𝑦 w − 6𝑦 ˆ 𝜈 h + Σ ( 𝑦 v 1 − 𝑦 v − Σ * 𝜚 Σ @ = 1 Σ • = 1 𝜁 , ε − 0.8ε, where ε = 0.001 Δ𝑦 = 𝜁 ⁄ / h ℎ 11 Δ𝑦 = ℎ
2D case 𝑀×𝑀 = 2×2 , ℎ q = ℎ ‹ = 0.2 / / Σ @ = Σ • = Œ − 0.8ε , 𝑅 = ε Œ , ε = 0.01 12
Concluding remarks (-) , our • Unlike the asymptotic methodology, 𝜔 " ≈ ∑ -.• Ž 𝜁 - 𝜔 " analysis was based on local truncation error analysis , e.g., ∆q • (-) Ž 𝜔 ",lm/ = ∑ -.• -! 𝜔 " • However, Taylor expansion was done with respect to a “scaled” mesh ∆𝑦 = 𝜁 S ℎ. This trick has made LTE analysis applicable for optically thick mesh! • Our analysis has shown 𝑚 = 1/𝑙 , where 𝑙 is the spatial order of accuracy of an upwind difference scheme. The result is sharp. • This work has filled the theoretical gap posed by Larsen et al. thirty years ago. • In addition, it is worth mentioning that in genera central schemes have the thick diffusion limit only for smooth solutions, but not for nonsmooth solutions. 13
References • 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. , (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 14
Thank you! 15
Recommend
More recommend
