1. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Neutrino Transport in
Core Collapse Supernovae
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

2. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Supernova Classification
Spectroscopic Classification
Classification by Mechanism
Type Ia
No H
Thermonuclear Supernovae
(Death of White Dwarf Star)
No H
Type Ib
Type Ic
Type II
No H, No He
Core Collapse Supernovae
(Death of Massive Stars)
H and He Present
Type Id
No H, No He
Hypernovae
(a.k.a. Type Ic Hypernovae)
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

3. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Core Collapse Supernovae as Element Factories
Core collapse supernovae are the dominant source of the elements between oxygen and iron.
Believed to be responsible for half the elements heavier than iron.
Single most important source of the elements in the Universe.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

4. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Core Collapse Supernova Paradigm
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

5. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Anatomy of a Supernova
Radiatively Driven!
Need Boltzmann Solution
Need Angular Distribution
Need Spectrum
Need Neutrino Distribution
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

6. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Newtonian versus GR
A comparison of key radii in a Newtonian versus a general relativistic model (25 Solar Masses):
Bruenn, DeNisco, and Mezzacappa (2001)
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

7. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Boltzmann Equation for Spherical Symmetry
O(v/c)
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Anthony Mezzacappa (ORNL)
Computational Methods in Transport

8. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Neutrino Emissivities and Opacities
“Standard”
Couple neutrino and antineutrino
distributions for a given flavor.
Couple flavors.
Burrows, Reddy, and Thompson (2004), astro-ph/
Burrows and Thompson, in Stellar Collapse, ed. C. Fryer, Kluwer.
Horowitz (2002), Phys. Rev. D65,
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

9. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Conservation of Energy and Lepton Number
Present a significant challenge…
erg radiated in neutrinos
erg explosion energy
Will need to conserve total energy to
0.1% over the entire simulation of
cycles!
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

10. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Conservation of Energy and Lepton Number
Mean Lepton Fraction
Mean lepton fraction in the inner homologous
core sets the inner core mass scale.
Sets the initial shock energy, …
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

11. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Difficult to develop number- and energy-
conservative differencing for these “observer
corrections” (aberration, frequency shift).
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

12. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Lepton Number Conservation
Number is conserved in both the comoving and lab frames.
Generalized Emissivity and Opacity
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

13. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Lab Frame Energy Conservation
One can define a globally conserved total specific energy in the lab frame and express it
in terms of comoving frame moments.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

14. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Lab Frame Energy Conservation 1
Cancel
Cancel 2
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Anthony Mezzacappa (ORNL)
Computational Methods in Transport

15. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Lab Frame Energy Conservation
Adding the equations on the previous slide together, we obtain:
Globally conserved lab frame specific energy
in terms of comoving frame moments.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

16. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Origin of term 2 on slide 14.
Origin of term 1.
Differencing of the observer corrections (4th and 5th terms on LHS) is not independent of the
differencing chosen for the spatial and angular derivatives (second and third terms on LHS).
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

17. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Discrete version of
integration leading
to conservation of
energy equation, for
select terms.
Discrete version of term 2 on slide 14.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

18. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Seeking differencing for A.
Term 1 on slide 14.
Term 2.
Must cancel.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

19. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Differencing for A
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

20. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Differencing of Observer Corrections
Doppler Shift
Initially, operator split Doppler term:
“Lagrangian” variable in energy.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

21. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Equations (1) and (2) on previous slide have following finite difference representation:
The above equations uniquely define:
Number of neutrinos lost to group .
Number of neutrinos gained in group .
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

22. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Once
are known, we can express the distribution function update as:
Substituting, we obtain the finite differencing for the Doppler shift term in the B.E.:
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

23. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Differencing of Observer Corrections
Aberration
A similar procedure is used to derive the finite differencing of the angular aberration
term on the LHS of the Boltzmann equation.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

24. Anthony Mezzacappa (ORNL) Computational Methods in Transport
3D Boltzmann Equation
O(v/c)
Corresponding terms in the 3D,
O(v/c), Newtonian gravity case.
1D: 5 Terms
3D: 55 Terms
The proliferation of terms will present a unique challenge
to the procedures we have just outlined.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

25. Anthony Mezzacappa (ORNL) Computational Methods in Transport
See Cardall and Mezzacappa (2003) for definitions.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

26. Anthony Mezzacappa (ORNL) Computational Methods in Transport
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

27. Anthony Mezzacappa (ORNL) Computational Methods in Transport
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

28. Anthony Mezzacappa (ORNL) Computational Methods in Transport
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

29. Anthony Mezzacappa (ORNL) Computational Methods in Transport
General Case
M: 7D manifold of all trajectories of particles of mass m.
6-form “surface element” normal to phase flow on M.
Number of “occupied states” emerging from D.
Generalized Stoke’s Theorem:
6D closed hypersurface on M.
7D volume on M.
Number of occupied states emerging from D
is equal to the number of collisions in D.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

30. Anthony Mezzacappa (ORNL) Computational Methods in Transport
General Case
Liouville operator restricted to M.
If we express the exterior derivative as:
7-form volume element on M.
we obtain the general relativistic Boltzmann equation:
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

31. Anthony Mezzacappa (ORNL) Computational Methods in Transport
General Case
Instead, if we express the exterior derivative as:
where
New Variable
Reduces to F for spherically
symmetric, O(v/c) case.
Conservative formulation of
general relativistic kinetic theory.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

32. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Solving TeraScale Linear Systems
Correspondence between structure of integro-PDE and
underlying linear systems...
…Leads to Nonlinear Algebraic Equations
Linearize
Solve via Multi-D Newton-Raphson Method
Solve Large Sparse Linear Systems
Implicit Time Differencing…
Extremely Short Neutrino-Matter
Coupling Time Scales
Neutrino-Matter Equilibration
Neutrino Transport Time Scales
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

33. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Progress
"ADI" Preconditioner
D’Azevedo et al. (2004)
Split solution into dense block plus (tri-, penta-, septa-) diagonal solve.
Preconditioned A.
Preconditioned b.
Krylov Subspace:
Solution of
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

34. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Completed: Spherical Models with Boltzmann Transport
Newtonian
General Relativistic
Messer et al. (2002)
Liebendoerfer et al. (2002)
No Explosions!
New Microphysics?
High-Density Stellar Core Thermodynamics
Neutrino-Matter Interactions
New Macrophysics? (2D/3D Models)
Fluid Instabilities, Rotation, Magnetic Fields
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

35. Anthony Mezzacappa (ORNL) Computational Methods in Transport
2D Boltzmann Test Problem
Development of radiation field stationary state in a nonspherical fixed medium:
Density Distribution
Radiation Field Energy Density and Flux
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

36. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Validation: Comparison with MGFLD
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

37. Anthony Mezzacappa (ORNL) Computational Methods in Transport
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

38. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Validation: Comparison with MGVEF
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

39. Anthony Mezzacappa (ORNL) Computational Methods in Transport
Conclusions
The core collapse supernova problem presents a unique set of challenges to simulate
accurately radiation (neutrino) transport in stellar cores.
We have developed significant extensions of the discrete ordinates method to meet
these challenges. These have been successfully applied in 1D (spherically symmetric)
simulations.
Ongoing work in the 2D and 3D cases continue to stress our methods even further,
even at O(v/c).
Future Challenges
Generalization to full GR.
Quantum versus classical (Boltzmann) kinetics.
Neutrinos have mass!
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport

40. Anthony Mezzacappa (ORNL) Computational Methods in Transport
BIBLIOGRAPHY
Methods
Mezzacappa and Bruenn, Ap.J. 405, 669 (1993).
Mezzacappa and Messer, JCAM, 109, 281 (1999).
Liebendoerfer, Ph.D. Thesis, Univ. Basel (2000).
Messer, Ph.D. Thesis, Univ. of Tennessee (2001).
Liebendoerfer et al., Ap.J. Suppl (2004).
Cardall and Mezzacappa, Phys. Rev. D68, (2003).
Cardall, astro-ph/ (2004).
Implementation
Mezzacappa and Bruenn, Ap.J. 410, 740 (1993).
Mezzacappa et al., Phys. Rev. Lett. 86, 1935 (2001).
Liebendoerfer et al., Phys. Rev. D63, (2001).
Hix et al., Phys. Rev. Lett. 91, (2003).
Liebendoerfer et al., astro-ph/ (2003).
See poster by
Bronson Messer.
11/10/2018
Anthony Mezzacappa (ORNL)
Computational Methods in Transport