A M A Mod odifi fied ed S Step ep Ch Character eristic M - - PowerPoint PPT Presentation

A M A Mod

• difi

fied ed S Step ep Ch Character eristic M Method

• d

fo for Solving the SN 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

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Finite difference sweeping methods

Linear methods

Ø Step difference (SD)

• 1st-order upwind
• Positivity preserving
• Intermediate diffusion limit, ∆𝑦 = 𝜁%ℎ, where 𝑚 = 1

Ø Diamond difference (DD)

• 2nd-order
• Not positivity preserving
• Thick diffusion limit in interior homogeneous regions, 𝑚 = 0

Ø Step characteristic (SC)

• Weighted DD
• 2nd-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

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SC

4

,- ./ 𝜔1,345/6 − 𝜔1,385/6 + Σ;,3 58<-,/ 6

𝜔1,385/6 +

54<-,/ 6

𝜔1,345/6 =

=>,/ 6 ∑1@A5 B 58<-@,/ 6

𝜔1@,38C

D +

54<-@,/ 6

𝜔1@,345/6 𝑥1F +

G/ 6

where 𝛽1,3 =

54IJKL,/M//N- 58IJKL,/M//N- − 6,- =L,/./ = 54IJO//N- 58IJO//N- − 6,- P/

𝜐3 = Σ;,3ℎ3, cell optical thickness ℎ3 = mesh size of cell 𝑘

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𝜈1 ℎ3 𝜔1,345/6 − 𝜔1,385/6 + Σ;,3 1 − 𝛽1,3 2 𝜔1,385/6 + 1 + 𝛽1,3 2 𝜔1,345/6 = ΣU,3 2 V

1@A5 B

𝜔1@,3𝑥1F + 𝑅3 2 , 𝑘 = 1, … 𝑛, for 𝜈1 > 0

M-matrix and stability

In matrix form: 𝐁𝛀 = 𝐓 where A is lower diagonal matrix 𝐁 = ⋮ ⋮ ⋱ ⋮ ⋮ … − 𝜈1 ℎ3 + Σ;,3 1 − 𝛽1,3 2 𝜈1 ℎ3 + Σ;,3 1 + 𝛽1,3 2 … … ⋮ ⋮ ⋮ ⋱ ⋮ For A is a M-matrix, we need to have −

,- ./ + Σ;,3 58<-,/ 6

≤ 0, and therefore SD 𝛽1,3 = 1 : A is unconditionally M-matrix DD 𝛽1,3 = 0 : ℎ3 ≤

6,- dL,/ , or 𝜐3 ≤ 2𝜈1

SC: ℎ3 ≤

6,- dL,/ 58<-,/ , or 𝜐3 ≤ 6,- 58<-,/ = 5 58

C N- e/ fe//N-JC

𝜐3, and therefore A is unconditionally M-matrix > 1

Diffusion limit of SN – a recap

𝜈1 𝑒 𝑒𝑦 𝜔1 + Σ;𝜔1 = ΣU 2 V

1FA5 B

𝜔1F𝑥1F + 𝑅 2 Scaling 𝜯𝒖 → 𝜯𝒖

𝜻 ,

𝜯𝒃 → 𝜻𝜯𝒃 , 𝑹 → 𝜻𝑹 We have 𝜔1 =

n 6 + 𝑃 𝜁 , for 𝜁 → 0

− 𝑒 𝑒𝑦 1 3Σq 𝑒 𝑒𝑦 𝜚 + Σs𝜚 = 𝑅 Where 𝜚 satisfies the following diffusion equation

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𝜈1 𝑒 𝑒𝑦 𝜔1 + 𝛵; 𝜁 𝜔1 = 1 2 𝛵; 𝜁 − 𝜁𝛵u V

1FA5 B

𝜔1F𝑥1F + 𝜁𝑅 2

Diffusion limit of SC (Wang, 2019)

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𝛽1,3 = 1 + 𝑓

8 =L/ w x wy./ ,-

1 − 𝑓

8 =L/ w ⁄ wy./ ,-

− 2𝜈1 Σ;3 𝜁 𝜁%ℎ3 = 1 + 𝑓8 =L/

⁄ ./ ,- wyJC

1 − 𝑓8 =L/

⁄ ./ ,- wyJC −

2𝜈1 Σ;3ℎ3 𝜁%85 Proof (by contradiction).

• If 𝑚 > 1, then 𝛽1,3 ↓ 0 as 𝜁 ↓ 0, and thus SC tends to DD, whereas DD has 𝑚 = 0.
• If 𝑚 < 1, then 𝛽1,3 ↑ 1 for 𝜈~ > 0, and 𝛽1,3 ↓ −1 for 𝜈1 < 0, as 𝜁 ↓ 0. Thus, SC tends

to SD, but SD has 𝑚 = 1.

• So we should have 𝑚 = 1 for SC, and then 𝛽1,3 =

54IJ KL/M//N- 58IJ KL/M//N- − 6,- =L/./

.

Modified SC (mSC)

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𝛽1,3 =

54I

JKL,/M/ CJ•/ € /N-

58I

JKL,/M/ CJ•/ € /N- −

6,- =L,/./ 58•/

€ =

54I

JO/ x CJ•/ € N-

58I

JO/ x CJ•/ € N- −

6,- P/ 58•/

€ ,

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

Diffusion limit of mSC (Wang, 2019)

9

𝛽1,3 = 1 + 𝑓

8 =L/ w x wy./ 58•/

€

,-

1 − 𝑓

8 =L/ w x wy./ 58•/

€

,-

− 2𝜈1 Σ;3 𝜁 𝜁%ℎ3 1 − 𝑑

3 …

= 1 + 𝑓8 =L/

⁄ ./ ,- wyJC 58•/

€

1 − 𝑓8 =L/

⁄ ./ ,- wyJC 58•/

€ −

2𝜈1 Σ;3ℎ3 𝜁%85 1 − 𝑑

3 …

Proof.

• 𝑑

3 = 1 − 𝜁6 =†/ =L/.

• As 𝜁 ↓ 0, the 𝜻𝒎8𝟐 𝟐 − 𝒅𝒌

𝜸 term tends to zero, and thus 𝛽 ↓ 0. As a result, the SC

reverts to the DD scheme, and therefore it can attain the thick diffusion limit as DD does.

How about positivity?

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For A is a M-matrix, we need to have 𝜐3 ≤

6,- 58<-,/ = 𝐷𝜐3 , where 𝐷 = 58•/

€

58

O/ x CJ•/ € N- f O/ x CJ•/ € N-JC

𝛽1,3 = 1 + 𝑓8P/

x 58•/

€

,-

1 − 𝑓8P/

x 58•/

€

,-

− 2𝜈1 τ3 1 − 𝑑

3 …

0.01 0.1 1 10 100 0.01 0.1 1 10 100

c=0 0.2 0.4 0.6 0.8 0.9 0.99

τ3/ 𝜈1 𝐷

Numerical results – accuracy

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𝑀 = 1 cm ℎ = 0.1 cm Σq = 5 cm85 𝑅 = 1 cm85 BC: Vacuum

Numerical Results – robustness

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Numerical Results – diffusion limit

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L = 1, h = 0.1, Σq =

5 ”,

Σ• =

5 ” − 0.8ε,

𝑅 = ε,

Conclusions

• We proposed a modified step characteristic method,

called mSC, which can improve the accuracy of the

• riginal 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

• riginal 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.

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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

SN 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 SN Neutron Transport Equation," Nucl. Sci. Eng., 193, 9, 982 (2019). https://doi.org/10.1080/00295639.2019.1582316

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Thank You!

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