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Multigrid discontinuous Galerkin method for multigroup particle transport Pablo Lucero Interdisciplinary Research Center for Scientific Computing Universit at Heidelberg DEAL.II Workshop 2015 Pablo Lucero (IWR) MGDG for multigroup particle

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Multigrid discontinuous Galerkin method for multigroup particle transport

Pablo Lucero

Interdisciplinary Research Center for Scientific Computing Universit¨ at Heidelberg

DEAL.II Workshop 2015

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Radiative transfer in astrophysics

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Radiative transfer in climatology

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Radiative transfer in neutron and gamma transport

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Particle density description

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Transport equation

Ω = (ψ, φ) Ψ = Ψ(Ω, E, x) Ψ′ = Ψ(Ω′, E′, x) σT = σT(E, x) σs(Ω′,E′) = σs(Ω′ → Ω, E′ → E, x) q = q(Ω, E, x) Transport equation Ω · ∇Ψ + σTΨ − Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′ = q, ∀(Ω, E, x) ∈ S × (0, Emax] × D Boundary condition Ψ(Ω, E) = 0 ∀(Ω, E) ∈ S × (0, Emax] × ∂D, Ω · n < 0

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Transport equation with fission

νσ′

f = νσf (E′)

χ = χ(E) Transport equation Ω · ∇Ψ + σTΨ − Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′ − χ Emax νσ′

f Ψ′dE′ = q

Eigenvalue problem Ω · ∇Ψ + σTΨ − Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′ = 1 keff χ Emax νσ′

f Ψ′dE′

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Diffusion approximation

J = J(E, x) =

  • S

ΩΨdΩ Φ = Φ(E, x) =

  • S

ΨdΩ Ω · ∇Ψ = ∇ · (ΩΨ) = ∇ · J J ≈ −D(E, x)∇Φ Diffusion equation −∇ · (D∇Φ) + σTΦ − Emax σs(E′)Φ′dE′ = q −∇ · (D∇Φ) + σTΦ − Emax σs(E′)Φ′dE′ − χ Emax νσ′

f Φ′dE′ = q

−∇ · (D∇Φ) + σTΦ − Emax σs(E′)Φ′dE′ = 1 keff χ Emax νσ′

f Φ′dE′

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Angle description

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Angle collocation Sn

Ω · ∇Ψ + σTΨ − Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′ = q, Ψi = Ψ(Ωi, E, x) Ψi′ = Ψ(Ωi′, E′, x) σi′i

s(E′) = σs(Ωi′ → Ωi, E′ → E, x)

qi = q(Ωi, E, x) Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′ ≈ Emax

n

  • i′=1

ωi′σi′i

s(E′)Ψi′dE′

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Angle collocation Sn

Ω · ∇Ψ + σTΨ − Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′ = q,                                    Ω1 · ∇Ψ1 + σTΨ1 − Emax

n

  • i′=1

ωi′σi′1

s(E′)Ψi′dE′ = q1

... Ωi · ∇Ψi + σTΨi − Emax

n

  • i′=1

ωi′σi′i

s(E′)Ψi′dE′ = qi

... Ωn · ∇Ψn + σTΨn − Emax

n

  • i′=1

ωi′σi′n

s(E′)Ψi′dE′ = qn

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Multigroup

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Multigroup

Ωi · ∇Ψi + σTΨi − Emax

n

  • i′=1

ωi′σi′i

s(E′)Ψi′dE′ = qi,

(0, Emax] = (0, E1] ∪ ... ∪ (Eg−1, Eg] ∪ ... ∪ (EG−1, Emax] Ψi,g = Eg

Eg−1

Ψ(Ωi, E, x)dE σi,g

T =

Eg

Eg−1

σT(E, x)dE Ψi,g σi′i,g′g

s

= Eg

Eg−1

Emax

n

  • i′=1

ωi′σi′i

s(E′)Ψi′dE′dE

Ψi,g′

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Multigroup

Ωi · ∇Ψi + σTΨi − Emax

n

  • i′=1

ωi′σi′i

s(E′)Ψi′,g′dE′ = qi,

                                       Ωi · ∇Ψi,1 + σT,1Ψi,1 −

G

  • g′=1

n

  • i′=1

ωi′σi′i,g′1

s

Ψi′,g′ = qi,1 ... Ωi · ∇Ψi,g + σT,gΨi,g −

G

  • g′=1

n

  • i′=1

ωi′σi′i,g′g

s

Ψi′,g′ = qi,g ... Ωi · ∇Ψi,G + σT,GΨi,G −

G

  • g′=1

n

  • i′=1

ωi′σi′i,g′G

s

Ψi′,g′ = qi,G

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Discrete angle and energy system

                                      1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

1,1

...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

g′,1

...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

G,1

... ... ... ... ...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

1,g

...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

g′,g

...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

G,g

... ... ... ... ...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

1,G

...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

g′,G

...       1, 1 ... i′, 1 ... n, 1 ... ... i′, 1 ... ... 1, i ... i′, i ... n, i ... ... i′, 1 ... ... 1, n ... i′, 1 ... n, n      

G,G

                               

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Discontinuous Galerkin finite elements

Ωi · ∇Ψi,g(x) + σT,g(x)Ψi,g(x) −

G

  • g′=1

n

  • i′=1

ωi′σi′i,g′g

s

(x)Ψi′,g′(x) = qi,g(x) Vh =

  • v ∈ L2(D)
  • v|K ∈ PK
  • {

{v} } := v1 + v2 2 { {vn} } := v1n1 + v2n2 2

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Discontinuous Galerkin finite elements

ah(ψ, v) =

  • K∈Th

G

  • g=1

n

  • i=1

ωi

  • K

 Ωi · ∇ψi,g + σT,gψi,g −

G

  • g′=1

n

  • i′=1

ωi′σi′i,g′g

s

ψi′,g′   vi,gdx +

  • F∈Fb

h

G

  • g=1
  • Ωi·n≤0

ωi

  • F

|Ωi · n| ψividx + bh(ψ, v) bh(ψ, v) =

  • F∈Fj

h

G

  • g=1

n

  • i=1

ωi

  • F
  • 4

max{4, σsh} |Ωi · n| { {ψi,gn} }{ {vi,gn} } − 2Ωi · { {ψi,gn} }{ {vi,g} }

  • dx

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Multigrid preconditioner with Schwarz smoothers

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Source iteration?

Coupled D=1e-05 1e-06 0.0001 0.01 1 100 10000 1e+06 Absorption 1e-06 0.0001 0.01 1 100 10000 1e+06 Frequency redistribution 10 20 30 40 50 Source Iteration D=1e-05 1e-06 0.0001 0.01 1 100 10000 1e+06 Absorption 1e-06 0.0001 0.01 1 100 10000 1e+06 Frequency redistribution 10 20 30 40 50

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Diffusion with σa = 1

−∇ · (ǫD∇Φ) + 1 ǫ

  • σTΦ −

Emax σs(E′)Φ′dE′

  • = q

lev|eps −1 −2 −3 −4 −5 −6 4 5 5 4 2 1 1 1 5 6 5 5 3 2 1 1 6 6 6 5 4 2 1 1 7 6 6 5 5 3 2 1 8 6 6 6 5 4 2 1 9 6 6 6 6 5 3 2

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Diffusion with σa = 0 2G

−∇ · (ǫD∇Φ) + 1 ǫ

  • σTΦ −

Emax σs(E′)Φ′dE′

  • = q

lev|eps −1 −2 −3 −4 −5 −6 4 5 5 4 2 1 1 1 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 6 8 6 6 6 6 6 6 6 9 6 6 6 6 6 6 6

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Diffusion with σa = 0 5G

−∇ · (ǫD∇Φ) + 1 ǫ

  • σTΦ −

Emax σs(E′)Φ′dE′

  • = q

lev|eps −1 −2 −3 −4 −5 −6 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 6 8 6 6 6 6 6 6 6 9 6 6 6 6 6 6 6

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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3 group transport

Ψ1(Ω1, x)

20 40 60 80 100 120 140

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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3 group transport

Ψ1(Ω2, x)

20 40 60 80 100 120 140

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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3 group transport

Ψ1(Ω3, x)

20 40 60 80 100 120 140

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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3 group transport

Ψ1(Ω4, x)

20 40 60 80 100 120 140

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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SLIDE 27

3 group transport

Φ1(x)

20 40 60 80 100 120

Result

20 40 60 80 100 120

Reference

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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SLIDE 28

3 group transport

Φ2(x)

10 20 30 40 50 60 70

Result

10 20 30 40 50 60 70

Reference

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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3 group transport

Φ3(x)

20 40 60 80 100 120 140

Result

20 40 60 80 100 120 140

Reference

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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3 group keff

Ω · ∇Ψ + σTΨ − Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′ = 1 keff χ Emax νσ′

f Ψ′dE′

#elem Result Reference diff [pcm] 16 0.9017874 0.9016819 12 32 0.9019407 0.9018984 5 64 0.9019622 0.9019520 2 128 0.9019651 0.9019654 < 1

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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1 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 7 7 8 8 9 5 3 5 8 9 9 10 11 6 3 5 9 10 10 11 12 7 3 4 8 10 11 11 12 8 3 4 8 10 11 11 12 9 3 4 8 9 10 11 12

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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2 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 10 10 11 12 13 5 3 5 10 11 12 13 14 6 3 5 10 12 13 14 15 7 3 4 10 12 13 14 15 8 3 4 10 12 13 14 15 9 3 4 9 11 12 13 15

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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3 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 12 12 13 14 15 5 3 5 12 14 15 16 17 6 3 5 12 14 15 17 18 7 3 4 12 14 15 17 18 8 3 4 11 14 15 16 18 9 3 4 11 13 15 16 17

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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4 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 12 14 15 16 17 5 3 5 13 15 17 18 20 6 3 5 13 16 17 19 20 7 3 4 12 16 17 19 20 8 3 4 12 15 17 18 20 9 3 4 11 15 17 18 19

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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5 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 13 15 16 18 19 5 3 5 14 17 19 20 22 6 3 5 14 18 19 21 22 7 3 4 13 17 19 21 22 8 3 4 12 17 19 20 22 9 3 4 12 16 18 20 21

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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SLIDE 36

6 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 14 16 18 19 20 5 3 5 15 19 20 22 24 6 3 5 14 19 21 23 24 7 3 4 14 19 21 22 24 8 3 4 13 18 20 22 24 9 3 4 12 17 20 21 23

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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SLIDE 37

7 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 14 17 19 20 21 5 3 5 15 20 22 23 25 6 3 5 14 21 22 24 26 7 3 4 14 20 22 24 26 8 3 4 13 19 21 23 25 9 3 4 13 18 21 23 25

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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SLIDE 38

8 group convergence

Ω · ∇Ψ + 1 ǫ

  • σTΨ −

Emax

  • S

σs(Ω′,E′)Ψ′dΩ′dE′

  • = q,

lev|eps −1 −2 −3 −4 −5 −6 4 3 5 14 18 19 21 22 5 3 5 15 21 23 25 27 6 3 5 15 22 23 26 28 7 3 4 14 21 23 25 28 8 3 4 13 20 23 25 27 9 − − 13 20 22 24 26

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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Conclusions

Energy iteration solver in each cell. Diffusion scales well in space, energy and parameter-wise. Transport does well in space and fair parameter-wise but does not scale well in energy = ⇒ multigrid in energy. ARPACK allows easy implementation of the eigenvalue problem with good performance.

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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SLIDE 40

To do

Prove the behavior of the diffusion solver, get insights for transport. Implement multigrid in energy (maybe multigrid in angle?) Use Schwarz-preconditioned Inexact Newton to include local thermodynamic equilibrium for astrophysics applications. Once a scalable algorithm is ready, implement the method matrix-free in parallel.

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

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SLIDE 41

Thank you.

Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015