Computational Methods for Neutrino Transport in Core-Collapse - - PowerPoint PPT Presentation

Computational Methods for Neutrino Transport in Core-Collapse Supernovae

Eirik Endeve endevee@ornl.gov March 22, 2017

Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 1 / 30

Outline

1

Background

2

Neutrino Transport Equations

3

Solving the Equations on a Computer

4

Some Examples

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Core-Collapse Supernovae (CCSNe)

Explosion of Massive Star (M 8 M⊙). Dominant Source of Heavy Elements.

Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 3 / 30

Computational Challenge

Computational models needed to interpret observations Neutrino transport most compute-intensive component of models

◮ Exascale computing challenge Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 4 / 30

Core-Collapse Supernovae (CCSNe)

Neutrinos Play Fundamental Role

Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 5 / 30

Core-Collapse Supernovae (CCSNe)

Neutrinos Play Fundamental Role

Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 6 / 30

Neutrino Mean-Free Path

Shock&Radius& Gain&Radius&

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Neutrino Transport: Boltzmann Equation

Stellar fluid semi-transparent to neutrinos in heating region Classical description based on non-negative distribution function dN = f (p, x, t) dp dx Kinetic equation: balance between advection and collisions L(f ) = C(f )

◮ Advection: Ballistic transport, relativistic effects ◮ Collisions: Emission/absorption, scattering, pair processes Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 8 / 30

Boltzmann Equation: Left-Hand Side

Phase-Space Advection

Relativistic Liouville operator L(f ) = pµ ∂f ∂xµ − pν pρ Γi

νρ

∂f ∂pi Neutrino four-momentum pµ = ε

• 1, cos ϑ, sin ϑ cos ϕ, sin ϑ sin ϕ

T Chirstoffel symbols Γµ

νρ = 1

2 g µσ ∂gσν ∂xρ + ∂gσρ ∂xν − ∂gνρ ∂xσ

• Astro-Particle Seminar, March 2017

Computational Methods for Neutrino Transport 9 / 30

Boltzmann Equation: Right-Hand Side

Neutrino-Matter Interactions

Electron capture e− + p ⇋ n + νe e− + (A, Z) ⇋ (A, Z − 1) + νe e+ + n ⇋ p + ¯ νe Scattering ν + α, A ⇋ α, A + ν ν + e−, e+, n, p ⇋ ν′ + (e−)′, (e+)′, n′, p′ Pair processes e− + e+ ⇋ ν + ¯ ν N + N ⇋ N′ + N′ + ν + ¯ ν

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Boltzmann Equation: Right-Hand Side

Integral Operators

Example: Neutrino-electron scattering C

• f
• (p) =
• 1 − f (p)

R3 R(p ← q) f (q) dq

− f (p)

• R3 R(p → q)
• 1 − f (q)
• dq

Computationally expensive to evaluate C

• f
• (pi) =

Np

• k=1

Mik(f ) f (pk) O(N2

p) operations

Must be evaluated for every x and t

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Solving the Equations

Challenges: High dimensionality f (p, x, t) ∈ R3 × R3 × R+

◮ High-order accurate methods

Multiple time scales τcol ≪ τadv

◮ Efficient time-integration methods

Robustness

◮ Distribution function bounded: f ∈ [0, 1] for Fermions Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 12 / 30

Model Equation

Consider Boltzmann equation in “slab symmetry” with simple collision term ∂tf + µ ∂xf = η − χ f f = f (x, t; ε, µ). Consider fixed ε ∈ R+ and µ = cos ϑ ∈ [−1, 1] η(x; ε) > 0 Emissivity χ(x; ε) > 0 Absorption opacity Collision term drives f towards equilibrium value fEq fEq = η/χ (= Fermi Dirac)

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Spatial Discretization

Divide space into N intervals Ii = {x : x ∈ [xi−1/2, xi+1/2]} ∀ i = 1, . . . , N

xi-1/2

xi+1/2 Δx

xi-1 xi xi+1

In each interval Ii, define the average ¯ fi(t) = 1 ∆x

• Ii

f (x, t) dx

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Spatial Discretization

Integrate Boltzmann equation over interval Ii ∂t ¯ fi = − 1 ∆x

• µf |i+1/2 − µf |i−1/2
• + ¯

ηi − 1 ∆x

• Ii

χ f dx (Exact Equation) Need to approximate µf |i+1/2 ≈ µf |i+1/2 = 1 2

• µ + |µ|

¯ fi + 1 2

• µ − |µ|

¯ fi+1 ¯ ηi ≈ ηi and 1 ∆x

• Ii

χ f dx ≈ χi ¯ fi So that ∂t ¯ fi = − 1 ∆x µf |i+1/2 − µf |i−1/2

• + ηi − χi ¯

fi = A(¯ fi−1, ¯ fi, ¯ fi+1) + C(¯ fi) = F(¯ fi−1, ¯ fi, ¯ fi+1)

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Spatial Discretization

Upwind Method

∂tf + µ ∂xf = 0 has solution f (x, t) = f0(x − µ t)

xi-1/2 xi+1/2

μ > 0 μ Δt

• µf |i+1/2 = 1

2

• µ + |µ|

¯ fi + 1 2

• µ − |µ|

¯ fi+1

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Time Integration

Divide time domain t0 < t1, t2, . . . , tn, tn+1, . . . , T

Define solution vector ¯ f (t) = (¯ f1(t), . . . , ¯ fN(t))T and write dt¯ f = F(¯ f ) Explicit ¯ f

n+1 = ¯

f

n +

tn+1

tn

F(¯ f (τ)) dτ ≈ ¯ f

n + ∆t F(¯

f

n)

(easy) Implicit ¯ f

n+1 = ¯

f

n +

tn+1

tn

F(¯ f (τ)) dτ ≈ ¯ f

n + ∆t F(¯

f

n+1)

(hard)

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Time Integration

Restrictions on the Time Step ∆t

Assume fEq,i, ¯ f n

i ∈ [0, 1]

Explicit method for collision term: ¯ f n+1

i

= (∆t χi) fEq,i + (1 − ∆t χi) ¯ f n

i

Need ∆t ≤ 1/χi for ¯ f n+1

i

∈ [0, 1] (not practical) Implicit method for collision term: ¯ f n+1

i

=

• ∆t χi

1 + ∆t χi

• fEq,i +
• 1

1 + ∆t χi

• ¯

f n

i

¯ f n+1

i

∈ [0, 1] for any ∆t ≥ 0

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Time Integration

Use combination of Explicit and Implicit methods

dt ¯ fi = A(¯ fi−1, ¯ fi, ¯ fi+1)

• Explicit

+ C(¯ fi)

• Implicit

=

+

I : ¯ f ⋆

i = ¯

f n

i + ∆t A(¯

f n

i−1, ¯

f n

i , ¯

f n

i+1)

II : ¯ f n+1

i

= ¯ f ⋆

i + ∆t C(¯

f n+1

i

)

Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 19 / 30

Bound-Preserving Spatial Discretization

Need to preserve f ∈ [0, 1] in advection step

Set of admissible states R = { f | f ≥ 0 and f ≤ 1} (convex set) Explicit advection step (λ = ∆t/∆x) ¯ f ⋆

i = ¯

f n

i − λ

• µf |i+1/2 −

µf |i−1/2

• = 1

2 λ

• |µ| + µ

¯ f n

i−1 +

• 1 − λ |µ|

¯ f n

i + 1

2 λ

• |µ| − µ

¯ f n

i+1

=

1

• k=−1

αk ¯ f n

i+k

where

1

• k=−1

αk = 1 For αk ≥ 0, ¯ f ⋆

i

is a convex combination of {¯ f n

i−1, ¯

f n

i , ¯

f n

i+1}

Need: ∆t ≤ ∆x |µ| (acceptable)

Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 20 / 30

Numerical Examples

Journal of Computational Physics 287 (2015) 151–183

Contents lists available at ScienceDirect

Journal of Computational Physics

www.elsevier.com/locate/jcp

Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates ✩

Eirik Endeve a,c,∗, Cory D. Hauck a,b, Yulong Xing a,b, Anthony Mezzacappa c

High-order method Local expansion: f (p, x, t) =

• k

ˆ fk(t) φk(p, x) Same principles (but somewhat more intricate)

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Advection Test with Smooth Analytical Solution

High-order methods can offer substantial savings in computational cost

10

2

10

4

10

6

10

8

10

−10

10

−5

10 Degrees of Freedom L1 Error Norm DG(1) DG(2) DG(3)

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Numerical Examples in Spherical Symmetry

ds2 = −α2 dt2 + ψ4 (dr2 + r2 dθ2 + r2 sin2 θ dφ2); f = f (r, µ, ε, t)

Boltzmann equation with relativistic gravity 1 α ∂f ∂t + 1 α ψ6 r 2 ∂ ∂r

• α ψ4 r 2 µ f
• Spatial advection

− 1 ε2 ∂ ∂ε

• ε3

1 ψ2 α ∂α ∂r µ f

• Energy advection

+ ∂ ∂µ 1 − µ2 ψ−2 1 r + 1 ψ2 ∂ψ2 ∂r − 1 α ∂α ∂r

• f
• Angular advection

= 0 Schwarzschild metric α = 1 − M

2 r

1 + M

2 r

and ψ = 1+ M 2 r

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Radiating Sphere Test

Neutrinos propagating out of gravitational well

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Radiating Sphere Test

Neutrinos propagating out of gravitational well

M"="0.0" E""="0.5"

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Varying the Mass M: f (r, µ, ε = const., tend)

M ¡= ¡0.0 ¡ E ¡ ¡= ¡0.5 ¡ M ¡= ¡2/3 ¡ E ¡ ¡= ¡0.5 ¡ M ¡= ¡2/3 ¡ E ¡ ¡= ¡0.3 ¡ M ¡= ¡0.2 ¡ E ¡ ¡= ¡0.5 ¡

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Neutrinos Streaming Out of Gravitational Well

Gravitational Redshift

Fermi-­‑Dirac ¡

First ¡Order ¡ Second ¡Order ¡ Third ¡Order ¡ Emi5ed ¡Spectrum, ¡r=1 ¡ Spectra, ¡r=3 ¡

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Positivity: f (r, µ, ε, t) and 1 − f (r, µ, ε, t)

Fermi-­‑Dirac ¡Spectrum ¡at ¡r=3 ¡ Without ¡limiter: ¡f ¡and ¡1-­‑f ¡< ¡0 ¡near ¡Fermi ¡surface ¡

Standard ¡Scheme ¡ BP ¡Scheme ¡ Standard ¡Scheme ¡ BP ¡Scheme ¡ Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 27 / 30

Homogeneous Sphere

Smit et al. 1997, A&A, 325, 203-211

Number Density Radius Number Density Radius ε = 10-1 ε = 10-8 Vacuum Vacuum

3rd Order 3rd Order

Transport tests

Optically thin limit Optically thick limit

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Summary

High dimensionality f (p, x, t) ∈ R3 × R3 × R+

◮ High-order accurate methods

Multiple time scales τcol ≪ τadv

◮ Efficient time-integration methods

Robustness

◮ Distribution function bounded: f ∈ [0, 1] for Fermions

There is a lot more to do!

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The End

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