So Solving g the S N Ne Neutron on Transpor ort Equation on Us Using H High O Order L Lax-Fr Friedrich chs W WENO Fa Fast Sweeping Methods Dean Wang, Tseelmaa Byambaakhuu The Ohio State University Sebastian Schunert Idaho National Laboratory Zeyun Wu Virginia Commonwealth University M&C 2019, Portland, Oregon, USA August 25-29, 2019
Outline • Background and motivation • Robustness • High order • Efficiency • LF-WENO methods • Theory (Wang 2019) • Numerical properties (Wang 2019, NSE) • Diffusion limit • Conclusion 2
Numerical methods for S N • Finite difference sweeping methods • SD: 1 st -order upwind; positivity preserving • DD: 2 nd -order; not positivity preserving • SC: weighted DD; 2 nd -order; positivity preserving; less accurate than DD for diffusive problems • Short characteristic methods • SC: constant source • LC: linear source & linear incoming flux; positivity preserving? • QC : Quadratic source & quadratic incoming flux; can be made to be positivity preserving (on-going work) • Galerkin methods: LD, FEM, DFEM • High-order • FEM or DFEM can be very robust with stabilization; however may not as efficient as finite difference sweeping methods. 3
Motivation • A sweeping based numerical method is more accurate than DD, and much more robust as well. • A challenging task… • Chen et al. in 2013 proposed Lax-Friedrichs fast sweeping methods for steady-state hyperbolic conservation laws. • A perfect framework for the S N transport equation! 4
S N in 2-D conservative form 𝑔 𝜔 # + 𝜔 & + Σ ( 𝜔 = 𝑡 𝜔, 𝑦, 𝑧 where, 𝑔 𝜔 = 𝜈𝜔 , 𝜔 = 𝜃𝜔 , 𝑡 𝜔, 𝑦, 𝑧 = 0 1 2 𝜚 𝑦, 𝑧 + 4 2 𝑅 𝑦, 𝑧 5
Finite difference discretization ; >,? − ; 𝑔 <=4 𝑔 <@4 <,?=4 B − B <,?@4 >,? > > + + Σ ( 𝜔 <,? = 𝑡 𝜔 <,? , 𝑦 < , 𝑧 ? (1) ∆𝑦 ∆𝑧 𝑗, 𝑘 + 1 2 Where 𝑗 − 1 𝑗 + 1 2 , 𝑘 2 , 𝑘 𝑗, 𝑘 ; 𝑔 <± D E ,? and B <,?± D E are numerical fluxes 𝑗, 𝑘 − 1 2 6
High order WENO fluxes For 2𝐿 − 1 – th order WENO scheme, the 𝐿 numerical fluxes are computed as Q@4 𝑑 LN 𝑔 ; L (2) 𝑔 = ∑ NOP 𝑠 = 0, … , 𝐿 − 1 , <@L=N,? , <= D E ,? which corresponds to 𝐿 different stencils: 𝑇 L 𝑗 = 𝑦 <@L , 𝑧 ? , … , 𝑦 <@L=Q@4 , 𝑧 ? , 𝑠 = 0, … , 𝐿 − 1 . Each of these numerical fluxes is 𝑙– th order accurate. The 2𝐿 − 1 – th order WENO flux is a superposition of all these K numerical fluxes Q@4 ; 𝑥 N ; N 𝑔 <=4 >,? = X 𝑔 (3) <=4 >,? NOP Q@4 𝑥 N = 1 , and are defined as The nonlinear weights 𝑥 N satisfy 𝑥 N ≥ 0 , ∑ NOP [ \ b \ (4) 𝑥 N = 𝛽 N = _`D [ \ , c=d \ . ∑ \]^ 7
Third order WENO (WENO3) For 𝐿 = 2 , the 2 nd -order accurate numerical fluxes for 𝜈 > 0 are given as = 4 <,? + 4 = − 4 <@4,? + p P 4 ; ; 𝑔 > 𝑔 > 𝑔 𝑔 > 𝑔 > 𝑔 <=4,? , (5) <,? <= D <= D E ,? E ,? And the linear weights are given by 𝑒 P = > 𝑒 4 = 4 p , (6) p Smoothness indicators are given by > , > 𝛾 P = 𝜐 P 𝑔 <=4,? − 𝑔 𝛾 4 = 𝜐 4 𝑔 <,? − 𝑔 (7) <,? <@4,? where 𝜐 P = 𝑏 ∗ max 𝑏𝑐𝑡 Σ (<=4,? − Σ (<,? , 𝑏𝑐𝑡 Σ n<=4,? − Σ n<,? ∆𝑦 𝜐 4 = 𝑐 ∗ max 𝑏𝑐𝑡 Σ (<,? − Σ (<@4,? , 𝑏𝑐𝑡 Σ n<,? − Σ n<@4,? ∆𝑦 8
Lax-Friedrichs sweeping framework Define Lax–Friedrichs fluxes: ; vw ; E ,? = ; (8𝑏) 𝑔 <= D 𝑔 <= D E ,? + 𝜔 <=4,? − 𝜔 <,? , 𝑗 = 1, … , 𝑂 # > vz y <,?= D B E = B <,?= D E + 𝜔 <,?=4 − 𝜔 <,? , 𝑘 = 1, … , 𝑂 & (8𝑐) > Then we have E ,? = ; vw ; ; 𝑔 <= D 𝑔 <= D E ,? − 𝜔 <=4,? − 𝜔 <,? > vz E = y <,?= D B <,?= D B E − 𝜔 <,?=4 − 𝜔 <,? > 9
LF-WENO >,? − 𝜏𝜈 >,? + 𝜏𝜈 ; 𝜔 <=4,? − 𝜔 <,? − ; ; ; 𝑔 <=4 𝑔 <@4 𝜔 <,? − 𝜔 <@4,? 2 2 ∆𝑦 − 𝜏𝜃 + 𝜏𝜃 y 𝜔 <,?=4 − 𝜔 <,? − y <,?=4 B <,?@4 B 𝜔 <,? − 𝜔 <,?@4 2 2 > > + + Σ ( 𝜔 <,? ∆𝑧 = 𝑡 𝜔 <,? , 𝑦 < , 𝑧 ? n } ~,• ,# ~ ,& • ∆#@ ; E,• @ ; E,• @ ‚ƒ @ ‚… ∆† @ y @ y ; ; € ~•D € ~`D E } ~•D,• =} ~`D,• „ ~,••D B „ ~,•`D B E } ~,••D =} ~,•`D ∆‡ (9) 𝜔 <,? = E E v w=z ∆† =ˆ ‰ ∆# ∆‡ 10
Computing algorithm • Initialize 𝜔 <,? and 𝑇 <,? • While 𝑓 > etol Ž 1. for 𝑜 = 1: % sweeping in angle ( 𝜈 > 0, 𝜃 > 0 ) 2 for 𝑗 = 1: 𝑂𝑦 % sweeping in x for 𝑘 = 1: 𝑂𝑧 % sweeping in y • Calculate ; N N 𝑔 and B , 𝑙 = 1,2 % Eq (5) <± D <± D E ,? E ,? • Calculate 𝛾 P , 𝛾 4 % Eq (7) • Calculate 𝛽 N , 𝑥 N , 𝑙 = 1,2 % Eq (4) • Calculate ; E ,? and ; 𝑔 <± D 𝑔 <± D % Eq (5) E ,? • Calculate ; ; E ,? and y 𝑔 <± D <± D B % Eq (8) E ,? • Calculate 𝜔 <,? % Eq (9) • Calculate 𝑇 <,? Ž Ž 2. for 𝑜 = 2 + 1: % sweeping in angle ( 𝜈 < 0, 𝜃 > 0 ) > … Ž pŽ 3. for 𝑜 = > + 1: % sweeping in angle ( 𝜈 < 0, 𝜃 < 0 ) 2 … pŽ 4. for 𝑜 = 2 + 1: 𝑂 % sweeping in angle ( 𝜈 > 0, 𝜃 < 0 ) 11
Spatial convergence 12
Manufactured solution 𝜔 𝑦, 𝑧, 𝜈 N , 𝜃 N = 𝑦 p 𝑧 p 2 − 𝑦 p 2 − 𝑧 p 24𝑦 > − 48𝑦 p + 30𝑦 2 − 6𝑦 ’ 𝑧 p 2 − 𝑧 p 𝜈 N 𝑅 N 𝑦, 𝑧 = 4 − Σ “ 𝜚 𝑦, 𝑧 +𝑦 p 2 − 𝑦 p 24𝑧 > − 48𝑧 p + 30𝑧 2 − 6𝑧 ’ 𝜃 N 13
Sweeping convergence rate Σ ( = 1 cm @4 and c =0.6 Σ ( = 5 cm @4 and c =0.6 14
Computational complexity * Computational complexity: the number of grid points x the number of iterations 15
Positivity? Note that LF-WENO3 can be rendered to be positivity preserving using the linear scaling limiter proposed by Zhang and Shu (2010). 16
Diffusion limit of S N 𝑒𝑦 𝜔 • + Σ ( 𝜔 • = Σ n 𝑒 2 𝜚 + 𝑅 𝜈 • 2 0 ‰ Σ ( → Σ “ → 𝜁Σ “ , 𝑅 → 𝜁𝑅 , Scaling — , ™ We have 𝜔 • = > + 𝑃 𝜁 , for 𝜁 → 0 Where 𝜚 satisfies the following diffusion equation − 𝑒 1 𝑒 𝑒𝑦 𝜚 + Σ œ 𝜚 = 𝑅 𝑒𝑦 3Σ › 17
Diffusion limit – smooth solution 4 4 Σ › = Σ ¡ = − 0.8ε , 𝑅 = ε , 𝑀 = 1, ℎ = 0.1 18
Diffusion limit – nonsmooth solution with boundary layer DD LF-WENO3 𝜁 = 0.01 19
Diffusion limit – 2D L×L = 2×2 , h # = h & = 0.2 4 4 Σ › = Σ ¡ = − 0.8ε , 𝑅 = ε , 20
A theoretical result on diffusion limit (Wang 2019, NSE): Δ𝑦 = 𝜁 ¦ ℎ = 𝜁 ⁄ 4 N ℎ 𝑚 = 0: Thick diffusion limit Larsen et al. 1987: 𝑚 = 1: Intermidiate diffusion limit Δ𝑦 = 𝜁 ⁄ 4 N ℎ Δ𝑦 = ℎ 21
Conclusions • LF-WENO3 is a sweeping scheme based on the Lax– Friedrichs fluxes with the WENO reconstruction. • It can achieve better accuracy than DD, and more importantly it possesses good positivity-preserving property. • In addition, LF-WENO3 can achieve almost linear computational complexity with underrelaxation. • Finally, LF-WENO3 has the diffusion limit of 𝑚 = 1/3 , which lies between the thick diffusion regime ( 𝑚 = 0) and the intermediate regime (𝑚 = 1 ). 22
References • W. Chen, C.-S. Chou, and C.-Y. Kao, “Lax-Friedrichs Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws,” J. Comput. Phys. , 234, 452 (2013) • 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). • X. Zhang, C.-W. Shu, “On maximum-principle-satisfying high order schemes for scalar conservation laws,” J. Comput. Phys. , 229 (2010). • X. Zhang, C.-W. Shu, “On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes,” J. Comput. Phys. , 229 (2010). • D. Wang, "The Asymptotic Diffusion Limit of Numerical Schemes for the SN Transport Equation," Nucl. Sci. Eng. , (2019). • 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). 23
Thank You! 24
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