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 Background 1 Neutrino Transport Equations 2 Solving the Equations on a Computer 3 Some Examples 4 Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 2 / 30

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 Gain&Radius& Shock&Radius& Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 7 / 30

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 ) d p d x 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 ) = p µ ∂ f ∂ f ∂ x µ − p ν p ρ Γ i νρ ∂ p i Neutrino four-momentum � � T p µ = ε 1 , cos ϑ, sin ϑ cos ϕ, sin ϑ sin ϕ Chirstoffel symbols 2 g µσ � ∂ g σν � νρ = 1 ∂ x ρ + ∂ g σρ ∂ x ν − ∂ g νρ Γ µ ∂ x σ 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 + ν e e − + ( A , Z ) ⇋ ( A , Z − 1) + ν e e + + n ⇋ p + ¯ ν e 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 Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 10 / 30

Boltzmann Equation: Right-Hand Side Integral Operators Example: Neutrino-electron scattering � � � � � C ( p ) = 1 − f ( p ) R 3 R ( p ← q ) f ( q ) d q f � � � − f ( p ) R 3 R ( p → q ) 1 − f ( q ) d q Computationally expensive to evaluate N p � � � C ( p i ) = M ik ( f ) f ( p k ) f k =1 O ( N 2 p ) operations Must be evaluated for every x and t Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 11 / 30

Solving the Equations Challenges: High dimensionality f ( p , x , t ) ∈ R 3 × R 3 × R + ◮ High-order accurate methods Multiple time scales τ col ≪ τ adv ◮ 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 ∂ t f + µ ∂ x f = 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 f Eq f Eq = η/χ (= Fermi Dirac) Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 13 / 30

Spatial Discretization Divide space into N intervals I i = { x : x ∈ [ x i − 1 / 2 , x i +1 / 2 ] } ∀ i = 1 , . . . , N x i-1 x i x i+1 Δx x i-1/2 x i+1/2 In each interval I i , define the average � 1 ¯ f i ( t ) = f ( x , t ) dx ∆ x I i Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 14 / 30

Spatial Discretization Integrate Boltzmann equation over interval I i � � � f i = − 1 η i − 1 ∂ t ¯ µ f | i +1 / 2 − µ f | i − 1 / 2 + ¯ χ f dx (Exact Equation) ∆ x ∆ x I i Need to approximate � � ¯ � � ¯ µ f | i +1 / 2 = 1 f i + 1 µ f | i +1 / 2 ≈ � µ + | µ | µ − | µ | f i +1 2 2 � 1 χ f dx ≈ χ i ¯ η i ≈ η i ¯ and f i ∆ x I i So that � � � f i = − 1 µ f | i +1 / 2 − � ∂ t ¯ + η i − χ i ¯ µ f | i − 1 / 2 f i ∆ x = A (¯ f i − 1 , ¯ f i , ¯ f i +1 ) + C (¯ f i ) = F (¯ f i − 1 , ¯ f i , ¯ f i +1 ) Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 15 / 30

Spatial Discretization Upwind Method ∂ t f + µ ∂ x f = 0 has solution f ( x , t ) = f 0 ( x − µ t ) μ > 0 μ Δt x i-1/2 x i+1/2 � � ¯ � � ¯ µ f | i +1 / 2 = 1 f i + 1 � µ + | µ | µ − | µ | f i +1 2 2 Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 16 / 30

Time Integration Divide time domain t 0 < t 1 , t 2 , . . . , t n , t n +1 , . . . , T Define solution vector ¯ f N ( t )) T and write d t ¯ f = F (¯ f ( t ) = (¯ f 1 ( t ) , . . . , ¯ f ) Implicit Explicit � t n +1 � t n +1 n +1 = ¯ n + n +1 = ¯ n + ¯ F (¯ ¯ F (¯ f ( τ )) d τ f f f ( τ )) d τ f f t n t n n + ∆ t F (¯ n + ∆ t F (¯ n ) n +1 ) ≈ ¯ ≈ ¯ f f (easy) (hard) f f Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 17 / 30

Time Integration Restrictions on the Time Step ∆ t Assume f Eq , i , ¯ f n i ∈ [0 , 1] Explicit method for collision term: ¯ = (∆ t χ i ) f Eq , i + (1 − ∆ t χ i ) ¯ f n +1 f n i i Need ∆ t ≤ 1 /χ i for ¯ f n +1 ∈ [0 , 1] (not practical) i Implicit method for collision term: � � � � ∆ t χ i 1 f n +1 ¯ ¯ f n = f Eq , i + i i 1 + ∆ t χ i 1 + ∆ t χ i ¯ f n +1 ∈ [0 , 1] for any ∆ t ≥ 0 i Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 18 / 30

Time Integration Use combination of Explicit and Implicit methods d t ¯ f i = A (¯ f i − 1 , ¯ f i , ¯ + C (¯ f i +1 ) f i ) � �� � ���� Explicit Implicit + = f ⋆ ¯ i = ¯ i + ∆ t A (¯ i − 1 , ¯ i , ¯ f n +1 ¯ = ¯ f ⋆ i + ∆ t C (¯ f n +1 f n f n f n f n I : i +1 ) II : ) i 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 ) � � ¯ i = ¯ µ f | i +1 / 2 − � � f ⋆ f n i − λ µ f | i − 1 / 2 � � ¯ � � ¯ � � ¯ = 1 i + 1 f n f n f n 2 λ | µ | + µ i − 1 + 1 − λ | µ | 2 λ | µ | − µ i +1 1 1 � � α k ¯ f n = where α k = 1 i + k k = − 1 k = − 1 For α k ≥ 0, ¯ is a convex combination of { ¯ i − 1 , ¯ i , ¯ f ⋆ f n f n f n i +1 } i ∆ t ≤ ∆ x Need: (acceptable) | µ | 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 � ˆ Local expansion: f ( p , x , t ) = f k ( t ) φ k ( p , x ) k Same principles (but somewhat more intricate) Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 21 / 30

Advection Test with Smooth Analytical Solution High-order methods can offer substantial savings in computational cost 0 10 L 1 Error Norm − 5 10 DG(1) DG(2) − 10 10 DG(3) 2 4 6 8 10 10 10 10 Degrees of Freedom Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 22 / 30

Numerical Examples in Spherical Symmetry ds 2 = − α 2 dt 2 + ψ 4 ( dr 2 + r 2 d θ 2 + r 2 sin 2 θ d φ 2 ); f = f ( r , µ, ε, t ) Boltzmann equation with relativistic gravity � � � � 1 ∂ f 1 ∂ − 1 ∂ 1 ∂α α ψ 4 r 2 µ f ε 3 ∂ t + ∂ r µ f α ψ 6 r 2 ψ 2 α ε 2 α ∂ r ∂ε � �� � � �� � Spatial advection Energy advection � � ψ − 2 � 1 � � 1 − µ 2 � ∂ψ 2 + ∂ r + 1 ∂ r − 1 ∂α f = 0 ∂µ ψ 2 α ∂ r � �� � Angular advection Schwarzschild metric α = 1 − M ψ = 1+ M 2 r and 1 + M 2 r 2 r Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 23 / 30

Radiating Sphere Test Neutrinos propagating out of gravitational well Astro-Particle Seminar, March 2017 Computational Methods for Neutrino Transport 24 / 30