Time-dependent simplified P N approximation to the equations of radiative transfer Martin Frank a, * , Axel Klar a , Edward W. Larsen b , Shugo Yasuda c a TU Kaiserslautern, Fachbereich Mathematik, Erwin-Schro ¨ dinger-Str., 67663 Kaiserslautern, Germany b Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan 48109, USA c Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan Received 16 March 2007; received in revised form 16 July 2007; accepted 16 July 2007 Available online 28 July 2007 Abstract The steady-state simplified P N approximation to the radiative transport equation has been successfully applied to many problems involving radiation. This paper presents the derivation of time-dependent simplified P N (SP N ) equations (up to N = 3) via two different approaches. First, we use an asymptotic analysis, similar to the asymptotic derivation of the steady-state SP N equations. Second, we use an approach similar to the original derivation of the steady-state SP N equa- tions and we show that both approaches lead to similar results. Special focus is put on the well-posedness of the equations and the question whether it can be guaranteed that the solution satisfies the correct physical bounds. Several numerical test cases are shown, including an analytical benchmark due to Su and Olson [B. Su, G.L. Olson, An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium, Ann. Nucl. Energy 24 (1997) 1035–1055.]. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Radiative transfer; Asymptotic analysis; Simplified P N equations 1. Introduction The transport of charged and uncharged particles in scattering and absorbing media is challenging from both a theoretical and a computational point of view. The up-to seven-dimensional phase space (space, time, velocity) of the Boltzmann transport equation combined with the necessity of a fine resolution still poses major problems. Time-dependent radiative transfer plays a role in astrophysics (supernova explosions), in the inter- action of short-pulsed lasers with plasmas, or in LIDAR (light detection and ranging) technology, to name a few. In the past, many sophisticated discretization schemes and approximate models have been developed. But still in many fields, deterministic particle transport calculations are hardly used. 0021-9991/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2007.07.009 * Corresponding author. Tel.: +49 6312054486; fax: +49 6312054986. E-mail addresses: frank@mathematik.uni-kl.de (M. Frank), klar@itwm.fhg.de (A. Klar), edlarsen@umich.edu (E.W. Larsen), yasuda@kuaero.kyoto-u.ac.jp (S. Yasuda). Journal of Computational Physics 226 (2007) 2289–2305 www.elsevier.com/locate/jcp