Derivation of the Isotropic Diffusion Source Approximation (IDSA) for Supernova Neutrino Transport by Asymptotic Expansions


We present Chapman--Enskog and Hilbert expansions applied to the $\mathcal O(v/c)$ Boltzmann equation for the radiative transfer of neutrinos in core-collapse supernovae. Based on the Legendre expansion of the scattering kernel for the collision integral truncated after the second term, we derive the diffusion limit for the Boltzmann equation by truncation of Chapman--Enskog or Hilbert expansions with reaction and collision scaling. We also give asymptotically sharp results obtained by the use of an additional time scaling. The diffusion limit determines the diffusion source in the isotropic diffusion source approximation (IDSA) of Boltzmann's equation [M. Liebendörfer, S.C. Whitehouse, and T. Fischer, Astrophys. J., 698 (2009), pp. 1174--1190], [H. Berninger et al., ESAIM Proc. 38, 2012, pp. 163--182] for which the free streaming limit and the reaction limit serve as limiters. Here, we derive the reaction limit as well as the free streaming limit by truncation of Chapman--Enskog or Hilbert expansions using reaction and collision scaling as well as time scaling, respectively. Finally, we explain why limiters are a good choice for the definition of the source term in the IDSA.


  1. Boltzmann equation
  2. radiative transfer
  3. neutrino
  4. core-collapse supernova
  5. asymptotic expansion
  6. diffusion limit

MSC codes

  1. 35B40
  2. 35Q20
  3. 35Q85
  4. 82C70
  5. 85-08
  6. 85A25

Get full access to this article

View all available purchase options and get full access to this article.


Q.R. Ahmad et al., Direct evidence for neutrino flavor transformation from neutral-current interactions in the Sudbury neutrino observatory, Phys. Rev. Lett., 89 (2002), 011301.
J.L. Anderson and H.R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica, 74 (1974), pp. 466--488.
W.D. Arnett, Gravitational collapse and weak interactions, Canad. J. Phys., 44 (1966), pp. 2553--2594.
W. Baade and F. Zwicky, Supernovae and cosmic rays, Phys. Rev., 45 (1934), p. 138.
J.N. Bahcall and R.L. Sears, Solar neutrinos, Annu. Rev. Astron. Astr., 10 (1972), pp. 25--44.
C. Bardos, R. Santos, and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), pp. 617--649.
H. Berninger, E. Frénod, M.J. Gander, M. Liebendörfer, J. Michaud, and N. Vasset, A mathematical description of the IDSA for supernova neutrino transport, its discretization and a comparison with a finite volume scheme for Boltzmann's equation, in CEMRACS'11: Multiscale Coupling of Complex Models in Scientific Computing, ESAIM Proc. 38, 2012, pp. 163--182.
H.A. Bethe, Supernova mechanisms, Rev. Modern Phys., 62 (1990), pp. 801--866.
H.A. Bethe and J.R. Wilson, Revival of a stalled supernova shock by neutrino heating, Astrophys. J., 295 (1985), pp. 14--23.
P.L. Bhatnagar, E.P. Gross, and M. Krook, A model for collision processes in gases. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), pp. 511--525.
R.M. Bionta, G. Blewitt, C.B. Bratton, D. Casper, and A. Ciocio, Observation of a neutrino burst in coincidence with supernova $1987$a in the large Magellanic cloud, Phys. Rev. Lett., 58 (1987), pp. 1494--1496.
S.W. Bruenn, Stellar core collapse: Numerical model and infall epoch, Astrophys. J. Suppl. S., 58 (1985), pp. 771--841.
S.W. Bruenn, A. Mezzacappa, W.R. Hix, J.M. Blondin, P. Marronetti, O.E.B. Messer, C.J. Dirk, and S. Yoshida, 2D and 3D core-collapse supernovae simulation results obtained with the CHIMERA code, J. Phys. Conf. Ser., 180 (2010), 012018.
R. Buras, H.-T. Janka, M.T. Keil, G.G. Raffelt, and M. Rampp, Electron neutrino pair annihilation: A new source for muon and tau neutrinos in supernovae, Astrophys. J., 587 (2003), pp. 320--326.
C. Cercignani and G.M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Birkhäuser Verlag, 2002.
S. Chapman, On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. Roy. Soc. London, 216 (1916), pp. 279--341.
S. Chapman, On the kinetic theory of gas; Part II. A composite monatomic gas, diffusion, viscosity and thermal conduction, Phil. Trans. Roy. Soc. London, 217 (1917), pp. 118--192.
S.A. Colgate and R.H. White, The hydrodynamic behavior of supernovae explosions, Astrophys. J., 143 (1966), pp. 626--681.
C.L. Cowan, Jr., F. Reines, F.B. Harrison, H.W. Kruse, and A.D. McGuire, Detection of the free neutrino: A confirmation, Science, 124 (1956), pp. 103--104.
P. Degond and S. Jin, A smooth transition model between kinetic and diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 2671--2687.
P. Degond, S. Jin, and L. Mieussens, A smooth transition between kinetic and hydrodynamic equations, J. Comput. Phys., 209 (2005), pp. 665--694.
D. Enskog, Kinetische Theorie der Vorgänge in mäß ig verdünnten Gasen, Ph.D. thesis, Uppsala, 1917.
J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland, 1972.
T. Goudon and F. Poupaud, Approximation by homogenization and diffusion of kinetic equations, Comm. Partial Differential Equations, 26 (2001), pp. 537--570.
M. Herant, W. Benz, W.R. Hix, C.L. Fryer, and S.A. Colgate, Inside the supernova: A powerful convective engine, Astrophys. J., 435 (1994), pp. 339--361.
K. Hirata, T. Kajita, M. Koshiba, M. Nakahata, and Y. Oyama, Observation of a neutrino burst from the supernova siv1987a, Phys. Rev. Lett., 58 (1987), pp. 1490--1493.
D. Hupp, M. Mendoza, I. Bouras, S. Succi, and H.J. Herrmann, Relativistic lattice Boltzmann method for quark-gluon plasma simulations, Phys. Rev. D, 84 (2011), 125015.
H.-T. Janka and E. Müller, Neutrino heating, convection, and the mechanism of type-II supernova explosions, Astron. Astrophys., 306 (1996), pp. 167--198.
R.Y. Käppeli, Numerical Methods for $3$D Magneto-Rotational Core-Collapse Supernova Simulation with Jet Formation, Ph.D. thesis, Universität Basel, 2011.
M.T. Keil, Supernova Neutrino Spectra and Applications to Flavor Oscillations, Ph.D. thesis, Technische Universität München, 2003.
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, 2nd ed., Course Theoret. Phys. 6, Pergamon Press, London, 1987.
J. Lattimer and F. Swesty, A generalized equation of state for hot, dense matter, Nuclear Phys. A, 535 (1991), pp. 331--376.
E.J. Lentz, O.E.B. Messer, W.R. Hix, and S.W. Bruenn, Interplay of neutrino opacities in core-collapse supernova simulations, Astrophys. J., 760 (2012), 94.
M. Liebendörfer, O.E.B. Messer, A. Mezzacappa, S.W. Bruenn, C.Y. Cardall, and F.-K. Thielemann, A finite difference representation of neutrino radiation hydrodynamics in spherically symmetric general relativistic spacetime, Astrophys. J. Suppl. S., 150 (2004), pp. 263--316.
M. Liebendörfer, A. Mezzacappa, F.-K. Thielemann, O.E.B. Messer, W.R. Hix, and S.W. Bruenn, Probing the gravitational well: No supernova explosion in spherical symmetry with general relativistic Boltzmann neutrino transport, Phys. Rev. D, 63 (2001), 103004.
M. Liebendörfer, M. Rampp, H.-T. Janka, and A. Mezzacappa, Supernova simulations with Boltzmann neutrino transport: A comparison of methods, Astrophys. J., 620 (2005), pp. 840--860.
M. Liebendörfer, S.C. Whitehouse, and T. Fischer, The isotropic diffusion source approximation for supernova neutrino transport, Astrophys. J., 698 (2009), pp. 1174--1190.
A. Marek, H. Dimmelmeier, H.-T. Janka, E. Müller, and R. Buras, Exploring the relativistic regime with Newtonian hydrodynamics: An improved effective gravitational potential for supernova simulations, Astron. Astrophys., 445 (2006), pp. 273--289.
A. Marek and H.-T. Janka, Delayed neutrino-driven supernova explosions aided by the standing accretion-shock instability, Astrophys. J., 694 (2009), pp. 664--696.
C. Marle, Modèle cinétique pour l'établissement des lois de la conduction de la chaleur et de la viscosité en théorie de la relativité, C. R. Acad. Sc. Paris, 260 (1965), pp. 6539--6541.
C. Marle, Sur l'établissement des équations de l'hydrodynamique des fluides relativites, I. L'équation de Boltzmann relativiste, Ann. Inst. H. Poincaré, 10 (1969), pp. 67--126.
C. Marle, Sur l'établissement des équations de l'hydrodynamique des fluides relativites, II. Méthodes de résolution approchée de l'équation de Boltzmann relativiste, Ann. Inst. H. Poincaré, 10 (1969), pp. 127--194.
A. Mezzacappa and S.W. Bruenn, A numerical method for solving the neutrino Boltzmann equation coupled to spherically symmetric stellar core collapse, Astrophys. J., 405 (1993), pp. 669--684.
A. Mezzacappa and S.W. Bruenn, Type II supernovae and Boltzmann neutrino transport: The infall phase, Astrophys. J., 405 (1993), pp. 637--668.
J. Michaud, Decomposition Techniques for the Radiative Transfer of Neutrinos in Core-Collapse Supernovae, Ph.D. thesis, Université de Genève, in preparation.
D. Mihalas and B. Weibel-Mihalas, Foundation of Radiation Hydrodynamics, Oxford University Press, New York, 1984.
B. Müller, H.-T. Janka, and H. Dimmelmeier, A new multi-dimensional general relativistic neutrino hydrodynamic code for core-collapse supernovae. I. Method and code tests in spherical symmetry, Astrophys. J. Suppl. S., 189 (2010), pp. 104--133.
E.S. Myra and S.A. Bludman, Neutrino transport and the prompt mechanism for type II supernovae, Astrophys. J., 340 (1989), pp. 384--395.
C.D. Ott, A. Burrows, L. Dessart, and E. Livne, Two-dimensional multiangle, multigroup neutrino radiation-hydrodynamic simulations of postbounce supernova cores, Astrophys. J., 685 (2008), pp. 1069--1088.
B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), pp. 205--244.
M. Rampp and H.-T. Janka, Spherically symmetric simulation with Boltzmann neutrino transport of core collapse and postbounce evolution of a 15 ${M}_\odot$ star, Astrophys. J., 539 (2000), pp. L33--L36.
M. Rampp and H.-T. Janka, Radiation hydrodynamics with neutrinos: Variable Eddington factor method for core-collapse supernova simulations, Astron. Astrophys., 396 (2002), pp. 361--392.
P. Romatschke, M. Mendoza, and S. Succi, Fully relativistic lattice Boltzmann algorithm, Phys. Rev. C, 84 (2011), 034903.
J. Speck and R.M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Comm. Math. Phys., 304 (2011), pp. 229--280.
K. Sumiyoshi, S. Yamada, H. Suzuki, and S. Chiba, Neutrino signals from the formation of a black hole: A probe of the equation of state of dense matter, Phys. Rev. Lett., 97 (2006), 091101.
T.A. Thompson, A. Burrows, and P.A. Pinto, Shock breakout in core-collapse supernovae and its neutrino signature, Astrophys. J., 592 (2003), pp. 434--456.
H. Trac and U.-L. Pen, A primer on Eulerian computational fluid dynamics for astrophysics, Publ. Astron. Soc. Pac., 115 (2003), pp. 303--321.
E.A. Uehling and G.E. Uhlenbeck, Transport phenomena in Einstein--Bose and Fermi--Dirac gases. I, Phys. Rev., 43 (1933), pp. 552--561.
P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), pp. 507--553.
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, UK, 1980.
J.R. Wilson and R.W. Mayle, Report on the progress of supernova research by the Livermore group, Phys. Rep., 227 (1993), pp. 97--111.
S.E. Woosley, The Great Supernova of 1987, Ann. N.Y. Acad. Sci., 571 (1989), pp. 397--413.
Z. Xu and C. Greiner, Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade, Phys. Rev. C, 71 (2005), 064901.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 3229 - 3265
ISSN (online): 1095-7154


Submitted: 24 September 2012
Accepted: 9 July 2013
Published online: 7 November 2013


  1. Boltzmann equation
  2. radiative transfer
  3. neutrino
  4. core-collapse supernova
  5. asymptotic expansion
  6. diffusion limit

MSC codes

  1. 35B40
  2. 35Q20
  3. 35Q85
  4. 82C70
  5. 85-08
  6. 85A25



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account