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An accurate treatment of scattering and diffusion in piecewise power-law models for cosmic ray and radiation/neutrino transport

Hopkins, Philip F. (2022) An accurate treatment of scattering and diffusion in piecewise power-law models for cosmic ray and radiation/neutrino transport. . (Unpublished)

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A popular numerical method to model the dynamics of a 'full spectrum' of cosmic rays (CRs), also applicable to radiation/neutrino hydrodynamics (RHD), is to discretize the spectrum at each location/cell as a piecewise power law in 'bins' of momentum (or frequency) space. This gives rise to a pair of conserved quantities (e.g. CR number and energy) which are exchanged between cells or bins, that in turn give the update to the normalization and slope of the spectrum in each bin. While these methods can be evolved exactly in momentum-space (considering injection, absorption, continuous losses/gains), numerical challenges arise dealing with spatial fluxes, if the scattering rates depend on momentum. This has often been treated by either by neglecting variation of those rates 'within the bin,' or sacrificing conservation -- introducing significant errors. Here, we derive a rigorous treatment of these terms, and show that the variation within the bin can be accounted for accurately with a simple set of scalar correction coefficients that can be written entirely in terms of other, explicitly-evolved 'bin-integrated' quantities. This eliminates the relevant errors without added computational cost, has no effect on the numerical stability of the method, and retains manifest conservation. We derive correction terms both for methods which explicitly integrate flux variables (two-moment or M1-like) methods, as well as single-moment (advection-diffusion, FLD-like) methods, and approximate corrections valid in various limits.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Hopkins, Philip F.0000-0003-3729-1684
Additional Information:Support for PFH was provided by NSF Research Grants 1911233 & 20009234, NSF CAREER grant 1455342, NASA grants 80NSSC18K0562, HST-AR-15800.001-A. Numerical calculations were run on the Caltech compute cluster “Wheeler,” allocations FTA-Hopkins supported by the NSF and TACC, and NASA HEC SMD-16-7592. Support for JS was provided by Rutherford Discovery Fellowship RDF-U001804 and Marsden Fund grant UOO1727, which are managed through the Royal Society Te Aparangi. DATA AVAILABILITY STATEMENT. The data supporting this article are available on reasonable request to the corresponding author.
Group:TAPIR, Astronomy Department
Funding AgencyGrant Number
Royal Society Te ApārangiRDF-U001804
Marsden Fund of the Royal Society of New ZealandUOO1727
Subject Keywords:cosmic rays — plasmas — methods: numerical — MHD — galaxies: evolution — ISM: structure
Record Number:CaltechAUTHORS:20220228-183322835
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113654
Deposited By: George Porter
Deposited On:28 Feb 2022 21:58
Last Modified:28 Feb 2022 21:58

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