# American Institute of Mathematical Sciences

October  2016, 9(5): 1351-1375. doi: 10.3934/dcdss.2016054

## The IDSA and the homogeneous sphere: Issues and possible improvements

 1 Université de Genève, Section de Mathématiques, 2-4, rue du Lièvre, CP 64, CH-1211 Genève, Switzerland

Received  December 2014 Revised  July 2015 Published  October 2016

In this paper, we are concerned with the study of the Isotropic Diffusion Source Approximation (IDSA) [6] of radiative transfer. After having recalled well-known limits of the radiative transfer equation, we present the IDSA and adapt it to the case of the homogeneous sphere. We then show that for this example the IDSA suffers from severe numerical difficulties. We argue that these difficulties originate in the min-max switch coupling mechanism used in the IDSA. To overcome this problem we reformulate the IDSA to avoid the problematic coupling. This allows us to access the modeling error of the IDSA for the homogeneous sphere test case. The IDSA is shown to overestimate the streaming component, hence we propose a new version of the IDSA, which is numerically shown to be more accurate than the old one. Analytical results and numerical tests are provided to support the accuracy of the new proposed approximation.
Citation: Jérôme Michaud. The IDSA and the homogeneous sphere: Issues and possible improvements. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1351-1375. doi: 10.3934/dcdss.2016054
##### References:
 [1] H. Berninger, E. Frénod, M. 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 ESAIM: Proc., CEMRACS'11: Multiscale Coupling of Complex Models in Scientific Computing, 38 (2012), 163-182. doi: 10.1051/proc/201238009. [2] H. Berninger, E. Frénod, M. Gander, M. Liebendörfer, J. Michaud and N. Vasset, Derivation of the IDSA for supernova neutrino transport by asymptotic expansions, SIMA, 45 (2013), 3229-3265. doi: 10.1137/12089243X. [3] S. Bruenn, Stellar core collapse: Numerical model and infall epoch, Astrophys. J. Suppl. S., 58 (1985), 771-841. doi: 10.1086/191056. [4] J. J. Duderstadt and W. R. Martin, Transport Theory, John Wiley & Sons, 1979. [5] C. Levermore and G. Pomraning, A flux-limited diffusion theory, The Astrophysical Journal, 248 (1981), 321-334. doi: 10.1086/159157. [6] M. Liebendörfer, S. Whitehouse and T. Fischer, The Isotropic Diffusion Source Approximation for Supernova Neutrino Transport, Astrophys. J., 698 (2009), 1174-1190. [7] J. Michaud, From Neutrino Radiative Transfer in Core-Collapse Supernovae to Fuzzy Domain Based Coupling Methods, PhD thesis, Université de Genève, 2015. [8] D. Mihalas and B. Weibel-Mihalas, Foundation of Radiation Hydrodynamics, Oxford University Press, 1984. [9] J. Smit, L. Van den Horn and S. Bludman, Closure in flux-limited neutrino diffusion and two-moment transport, Astronomy and Astrophysics, 356 (2000), 559-569.

show all references

##### References:
 [1] H. Berninger, E. Frénod, M. 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 ESAIM: Proc., CEMRACS'11: Multiscale Coupling of Complex Models in Scientific Computing, 38 (2012), 163-182. doi: 10.1051/proc/201238009. [2] H. Berninger, E. Frénod, M. Gander, M. Liebendörfer, J. Michaud and N. Vasset, Derivation of the IDSA for supernova neutrino transport by asymptotic expansions, SIMA, 45 (2013), 3229-3265. doi: 10.1137/12089243X. [3] S. Bruenn, Stellar core collapse: Numerical model and infall epoch, Astrophys. J. Suppl. S., 58 (1985), 771-841. doi: 10.1086/191056. [4] J. J. Duderstadt and W. R. Martin, Transport Theory, John Wiley & Sons, 1979. [5] C. Levermore and G. Pomraning, A flux-limited diffusion theory, The Astrophysical Journal, 248 (1981), 321-334. doi: 10.1086/159157. [6] M. Liebendörfer, S. Whitehouse and T. Fischer, The Isotropic Diffusion Source Approximation for Supernova Neutrino Transport, Astrophys. J., 698 (2009), 1174-1190. [7] J. Michaud, From Neutrino Radiative Transfer in Core-Collapse Supernovae to Fuzzy Domain Based Coupling Methods, PhD thesis, Université de Genève, 2015. [8] D. Mihalas and B. Weibel-Mihalas, Foundation of Radiation Hydrodynamics, Oxford University Press, 1984. [9] J. Smit, L. Van den Horn and S. Bludman, Closure in flux-limited neutrino diffusion and two-moment transport, Astronomy and Astrophysics, 356 (2000), 559-569.
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