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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

SLIDE 1

On a Recent Theoretical Result

n Diffusion Limits of Numerical Methods

for the SN Transport Equation in Optically Thick Diffusive Regimes

Dean Wang The Ohio State University

ICTT-26 Sorbonne University, Paris, France September 23-27, 2019

SLIDE 2

Outline

Background and motivation
Larsen et at., 1987
Asymptotic analysis
Our result (Wang, 2019)
Local truncation error analysis
Remarks

SLIDE 3

Diffusion limit of SN – a recap

𝜈" 𝑒 𝑒𝑦 𝜔" + Σ(𝜔" = Σ* 2 ,

𝜔-𝑥- + 𝑅 2 Scaling Σ( → 45

6 ,

Σ7 → 𝜁Σ7 , 𝑅 → 𝜁𝑅 We have 𝜔" =

9 : + 𝑃 𝜁 , for 𝜁 → 0

− 𝑒 𝑒𝑦 1 3Σ@ 𝑒 𝑒𝑦 𝜚 + ΣB𝜚 = 𝑅 Where 𝜚 satisfies the following diffusion equation

𝝂𝒏 𝒆 𝒆𝒚 𝝎𝒏 + 𝜯𝒖 𝜻 𝝎𝒏 = 𝟐 𝟑 𝜯𝒖 𝜻 − 𝜻𝜯𝒃 ,

𝒐.𝟐 𝑶

𝝎𝒐𝒙𝒐 + 𝜻𝑹 𝟑

SLIDE 4

Larsen et al.’s result (1987)

∆𝑦 = 𝜁Sℎ A mesh size to resolve variations in the solution: The three diffusion regimes are defined by

𝑚 = V thick 1 intermediate ≥ 2 thin

SLIDE 5

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

rigin (Δ𝑦, 𝜁) = (0, 0) tangent to the vertical axis.”

SLIDE 6

Our result (Wang, 2019)

Δ𝑦 = 𝜁 ⁄

/ hℎ

where 𝑙 is the order of accuracy of spatial discretization, and 𝑙 ≥ 1

𝑙 = 3 𝑙 = 7

SLIDE 7

How to … …

Lo Local al tr trunc uncatio tion n error analy analysis is

𝜈" 6∆𝑦l 2𝜔",lm/ + 3𝜔",l − 6𝜔",ln/ + 𝜔",ln: + Σ(l 𝜁 𝜔",l = 1 2 Σ(l 𝜁 − 𝜁Σ7l ,

𝜔-,l𝑥- + 𝜁𝑅l 2 𝜈" 𝑒 𝑒𝑦 𝜔" + Σ( 𝜁 𝜔" = 1 2 Σ( 𝜁 − 𝜁Σ7 ,

𝜔-𝑥- + 𝜁𝑅 2

3rd-order upwind method: Asymptotic SN:

SLIDE 8

Taylor expansion at the center of cell 𝑘 gives

𝜈"

p pq 𝜔",l + 45r 6 𝜔",l + 𝑩 = / : 45r 6 − 𝜁Σ7l ∑-./

𝜔-,l𝑥- +

6ur :

𝑩 = 𝜈" ∆𝑦l

12 𝜔",l

w + 𝑃

∆𝑦l

where Let ∆𝒚𝒌 = 𝜻𝒎𝒊𝒌, we have

𝑩 = 𝜈" 𝜁Sℎl

12 𝜔",l

w + 𝑃

𝜁Sℎl

In order for Eq. (1) to approach the diffusion solution, we need to have

(1)

𝑩 ≪ 𝜁𝑅l 2

SLIDE 9

𝜈" 𝜁Sℎl

12 𝜔",l

w ≪ 𝜁𝑅l

2 𝜁vS ≪ 𝜁 If the solution is sufficiently smooth, then 𝜔",l

w ≈ 𝑃 1 . In addition ℎl ≈

𝑃 1 , we have 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

SLIDE 10

Numerical results

LF-WENO3

SLIDE 11

Δ𝑦 = ℎ Δ𝑦 = 𝜁 ⁄

/ hℎ

Numerical results – manufactured solution

Σ@ = 1 𝜁 , Σ• = 1 ε − 0.8ε, where ε = 0.001

† 𝜔 𝑦, 𝜈h = 𝑦v 1 − 𝑦 v 𝑅 𝑦 = 2 3𝑦: − 12𝑦v + 15𝑦w − 6𝑦ˆ 𝜈h + Σ(𝑦v 1 − 𝑦 v − Σ*𝜚

SLIDE 12

2D case

𝑀×𝑀 = 2×2, ℎq = ℎ‹ = 0.2 Σ@ =

/ Œ,

Σ• =

/ Œ − 0.8ε,

𝑅 = ε

ε = 0.01

SLIDE 13

Concluding remarks

Unlike the asymptotic methodology, 𝜔" ≈ ∑-.•

𝜁-𝜔"

(-), our

analysis was based on local truncation error analysis, e.g.,

𝜔",lm/ = ∑-.•

Ž ∆q •

! 𝜔"

(-)

However, Taylor expansion was done with respect to a “scaled”

mesh ∆𝑦 = 𝜁Sℎ. This trick has made LTE analysis applicable for

ptically 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.

SLIDE 14

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 SN 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 SN Neutron Transport Equation," Nucl. Sci. Eng., 193, 9, 982 (2019).

https://doi.org/10.1080/00295639.2019.1582316

SLIDE 15

On a Recent Theoretical Result

for the SN Transport Equation in Optically Thick Diffusive Regimes

Dean Wang The Ohio State University

Outline

Diffusion limit of SN – a recap

Larsen et al.’s result (1987)

In IX. DISCUSSION, Larsen et al. propose

“However, other choices of 𝑚 are possible and may be of

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

Our result (Wang, 2019)

Δ𝑦 = 𝜁 ⁄

/ hℎ

How to … …

Lo Local al tr trunc uncatio tion n error analy analysis is

3rd-order upwind method: Asymptotic SN:

Taylor expansion at the center of cell 𝑘 gives

inf 𝑚 = 1/𝑙

Numerical results

Numerical results – manufactured solution

2D case

ε = 0.01

Concluding remarks

analysis was based on local truncation error analysis, e.g.,

mesh ∆𝑦 = 𝜁Sℎ. This trick has made LTE analysis applicable for

accuracy of an upwind difference scheme. The result is sharp.

thirty years ago.

schemes have the thick diffusion limit only for smooth solutions, but not for nonsmooth solutions.

References

Solutions of Numerical Transport Problems in Optically Thick, Diffusive Regimes,” J. Comput. Phys., 69, 283 (1987).

Schemes for the SN Transport Equation," Nucl. Sci. Eng., (2019). https://doi.org/10.1080/00295639.2019.1638660

WENO Fast Sweeping Methods for the SN Neutron Transport Equation," Nucl. Sci. Eng., 193, 9, 982 (2019).

Thank you!