computational methods for neutrino transport in core-collapse … · 2017. 4. 6. · astro-particle...

Computational Methods for Neutrino Transportin Core-Collapse Supernovae

Eirik [email protected]

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


Core-Collapse Supernovae (CCSNe)Explosion of Massive Star (M & 8 M�). Dominant Source of Heavy Elements.


Computational Challenge

Computational models needed to interpret observations

Neutrino transport most compute-intensive component of models

I Exascale computing challenge


Core-Collapse Supernovae (CCSNe)Neutrinos Play Fundamental Role

40 H.-Th. Janka et al. / Physics Reports 442 (2007) 38–74

Fig. 1. Schematic representation of the evolutionary stages from stellar core collapse through the onset of the supernova explosion to the neutrino-drivenwind during the neutrino-cooling phase of the proto-neutron star (PNS). The panels display the dynamical conditions in their upper half, with arrowsrepresenting velocity vectors. The nuclear composition as well as the nuclear and weak processes are indicated in the lower half of each panel. Thehorizontal axis gives mass information. MCh means the Chandrasekhar mass and Mhc the mass of the subsonically collapsing, homologous innercore. The vertical axis shows corresponding radii, with RFe, Rs, Rg, Rns, and R! being the iron core radius, shock radius, gain radius, neutron starradius, and neutrinosphere, respectively. The PNS has maximum densities " above the saturation density of nuclear matter ("0).


Neutrino Mean-Free Path

Shock&Radius&Gain&Radius&


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 )

I Advection: Ballistic transport, relativistic effects

I Collisions: Emission/absorption, scattering, pair processes


Boltzmann Equation: Left-Hand SidePhase-Space Advection

Relativistic Liouville operator

L(f ) = pµ∂f

∂xµ− pν pρ Γi

νρ

∂f

∂pi

Neutrino four-momentum

pµ = ε(

1, cosϑ, sinϑ cosϕ, sinϑ sinϕ)T

Chirstoffel symbols

Γµνρ =1

2gµσ

( ∂gσν∂xρ

+∂gσρ∂xν

− ∂gνρ∂xσ

)


Boltzmann Equation: Right-Hand SideNeutrino-Matter Interactions

Electron capture

e− + p n + νe

e− + (A,Z ) (A,Z − 1) + νe

e+ + n p + ν̄e

Scattering

ν + α,A α,A + ν

ν + e−, e+, n, p ν′ + (e−)′, (e+)′, n′, p′

Pair processes

e− + e+ ν + ν̄

N + N N ′ + N ′ + ν + ν̄


Boltzmann Equation: Right-Hand SideIntegral 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(N2p) operations

Must be evaluated for every x and t


Solving the Equations

Challenges:

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

I High-order accurate methods

Multiple time scales τcol � τadv

I Efficient time-integration methods

Robustness

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


Model Equation

Consider Boltzmann equation in “slab symmetry” with simple collision term

∂t f + µ∂x f = η − χ f

f = f (x , t; ε, µ). Consider fixed ε ∈ R+ and µ = cosϑ ∈ [−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)


Spatial Discretization

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

xi-1/2 xi+1/2Δx

xi-1 xi xi+1

In each interval Ii , define the average

f̄i (t) =1

∆x

∫Ii

f (x , t) dx


Spatial Discretization

Integrate Boltzmann equation over interval Ii

∂t f̄i = − 1

∆x

(µf |i+1/2 − µf |i−1/2

)+ η̄i −

1

∆x

∫Ii

χ f dx (Exact Equation)

Need to approximate

µf |i+1/2 ≈ µ̂f |i+1/2 =1

2

(µ+ |µ|

)f̄i +

1

2

(µ− |µ|

)f̄i+1

η̄i ≈ ηi and1

∆x

∫Ii

χ f dx ≈ χi f̄i

So that

∂t f̄i = − 1

∆x

(µ̂f |i+1/2 − µ̂f |i−1/2

)+ ηi − χi f̄i

= A(f̄i−1, f̄i , f̄i+1) + C(f̄i ) = F(f̄i−1, f̄i , f̄i+1)


Spatial DiscretizationUpwind Method

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

xi-1/2 xi+1/2

μ>0

μΔt

µ̂f |i+1/2 =1

2

(µ+ |µ|

)f̄i +

1

2

(µ− |µ|

)f̄i+1


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

Define solution vector f̄ (t) = (f̄1(t), . . . , f̄N(t))T and write dt f̄ = F(f̄ )

Explicit

f̄n+1

= f̄n

+

∫ tn+1

tnF(f̄ (τ)) dτ

≈ f̄n

+ ∆tF(f̄n) (easy)

Implicit

f̄n+1

= f̄n

+

∫ tn+1

tnF(f̄ (τ)) dτ

≈ f̄n

+ ∆tF(f̄n+1

) (hard)


Time IntegrationRestrictions on the Time Step ∆t

AssumefEq,i , f̄

ni ∈ [0, 1]

Explicit method for collision term:

f̄ n+1i = (∆t χi ) fEq,i + (1−∆t χi ) f̄

ni

Need ∆t ≤ 1/χi for f̄ n+1i ∈ [0, 1] (not practical)

Implicit method for collision term:

f̄ n+1i =

( ∆t χi

1 + ∆t χi

)fEq,i +

( 1

1 + ∆t χi

)f̄ ni

f̄ n+1i ∈ [0, 1] for any ∆t ≥ 0


Time IntegrationUse combination of Explicit and Implicit methods

dt f̄i = A(f̄i−1, f̄i , f̄i+1)︸︷︷︸Explicit

+ C(f̄i )︸︷︷︸Implicit

= +

I : f̄ ?i = f̄ ni + ∆tA(f̄ ni−1, f̄ni , f̄

ni+1) II : f̄ n+1

i = f̄ ?i + ∆t C(f̄ n+1i )


Bound-Preserving Spatial DiscretizationNeed 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)

f̄ ?i = f̄ ni − λ(µ̂f |i+1/2 − µ̂f |i−1/2

)=

1

2λ(|µ|+ µ

)f̄ ni−1 +

(1− λ |µ|

)f̄ ni +

1

2λ(|µ| − µ

)f̄ ni+1

=1∑

k=−1

αk f̄ni+k where

1∑k=−1

αk = 1

For αk ≥ 0, f̄ ?i is a convex combination of {f̄ ni−1, f̄ni , f̄

ni+1}

Need: ∆t ≤ ∆x

|µ|(acceptable)


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

a Computational and Applied Mathematics Group, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USAb Department of Mathematics, University of Tennessee Knoxville, TN 37996-1320, USAc Department of Physics and Astronomy, University of Tennessee Knoxville, TN 37996-1200, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 October 2014Received in revised form 3 February 2015Accepted 5 February 2015Available online 11 February 2015

Keywords:Boltzmann equationRadiation transportHyperbolic conservation lawsDiscontinuous GalerkinMaximum principleHigh order accuracy

We extend the positivity-preserving method of Zhang and Shu [49] to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stability-preserving, Runge–Kutta (SSP-RK) time integration. Special care is taken to ensure that the method preserves strict bounds for the phase space distribution function f ; i.e., f ∈ [0, 1]. The combination of suitable CFL conditions and the use of the high-order limiter proposed in [49] is sufficient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergence-free property of the phase space flow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments — including one example in spherical symmetry adopting the Schwarzschild metric — demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle.

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we design discontinuous Galerkin methods for the solution of the collision-less, conservative Boltzmann equation in general curvilinear coordinates

✩ This research is sponsored, in part, by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory (ORNL), managed by UT-Battelle, LLC for the U.S. Department of Energy under Contract No. De-AC05-00OR22725. It used resources of the Oak Ridge Leadership Computing Facility at ORNL provided through the INCITE program and a Director’s Discretionary allocation. The research of the second author is supported in part by NSF under Grant No. 1217170. The research of the third author is supported in part by NSF grant DMS-1216454.

* Corresponding author. Tel.: +1 865 576 6349; fax: +1 865 241 0381.E-mail addresses: [email protected] (E. Endeve), [email protected] (C.D. Hauck), [email protected] (Y. Xing), [email protected] (A. Mezzacappa).

http://dx.doi.org/10.1016/j.jcp.2015.02.0050021-9991/© 2015 Elsevier Inc. All rights reserved.

High-order method

Local expansion: f (p, x , t) =∑k

f̂k(t)φk(p, x)

Same principles (but somewhat more intricate)


Advection Test with Smooth Analytical SolutionHigh-order methods can offer substantial savings in computational cost

102 104 106 108

10−10

10−5

100

Degrees of Freedom

L1 Erro

r Nor

m

DG(1)DG(2)DG(3)


Numerical Examples in Spherical Symmetryds2 = −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

αψ6 r2

∂

∂r

(αψ4 r2 µ f

)︸︷︷︸

Spatial advection

− 1

ε2

∂

∂ε

(ε3 1

ψ2 α

∂α

∂rµ 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 + M2 r

and ψ = 1+M

2 r


Radiating Sphere TestNeutrinos propagating out of gravitational well

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


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


Neutrinos Streaming Out of Gravitational WellGravitational Redshift

Fermi-‐Dirac

First Order Second Order Third Order

Emi5ed Spectrum, r=1

Spectra, r=3


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


Homogeneous SphereSmit et al. 1997, A&A, 325, 203-211

Num

ber

Den

sity

Radius

Num

ber

Den

sity

Radius

ε = 10-1 ε = 10-8 Vacuum Vacuum

3rd Order 3rd Order

Transport tests

Optically thin limit Optically thick limit


Summary

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

I High-order accurate methods

Multiple time scales τcol � τadv

I Efficient time-integration methods

Robustness

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

There is a lot more to do!


The End


computational methods for neutrino transport in core-collapse … · 2017. 4. 6. · astro-particle...

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