A linear stability analysis for nonlinear, grey, thermal radiative transfer problems Allan B. Wollaber a,⇑ , Edward W. Larsen b a CCS-2, Los Alamos National Laboratory, P.O. Box 1663, MS D-409, Los Alamos, NM 87545, United States b University of Michigan, 2355 Bonisteel Blvd., Ann Arbor, MI 48109, United States article info Article history: Received 2 September 2010 Received in revised form 10 November 2010 Accepted 13 November 2010 Available online 18 November 2010 Keywords: Monte Carlo Stability Monotonicity Radiative transfer abstract We present a new linear stability analysis of three time discretizations and Monte Carlo interpretations of the nonlinear, grey thermal radiative transfer (TRT) equations: the widely used ‘‘Implicit Monte Carlo’’ (IMC) equations, the Carter Forest (CF) equations, and the Ahrens–Larsen or ‘‘Semi-Analog Monte Carlo’’ (SMC) equations. Using a spatial Fourier analysis of the 1-D Implicit Monte Carlo (IMC) equations that are linearized about an equilibrium solution, we show that the IMC equations are unconditionally stable (undamped perturbations do not exist) if a, the IMC time-discretization parameter, satis- fies 0.5 < a 6 1. This is consistent with conventional wisdom. However, we also show that for sufficiently large time steps, unphysical damped oscillations can exist that correspond to the lowest-frequency Fourier modes. After numerically confirming this result, we develop a method to assess the stability of any time discretization of the 0-D, nonlinear, grey, thermal radiative transfer problem. Subsequent analyses of the CF and SMC methods then demonstrate that the CF method is unconditionally stable and monotonic, but the SMC method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium. This stability theory provides new conditions on the time step to guarantee monotonicity of the IMC solution, although they are likely too conservative to be used in practice. Theoretical predictions are tested and confirmed with numerical experiments. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The approximate Implicit Monte Carlo (IMC) equations for thermal radiation transport (TRT) are known to produce unphysical solutions when a sufficiently large time step is used [1–3], but little attention has been given to the theoretical stability properties of the IMC equations. Additionally, it is not known whether and to what extent two promising alterna- tives to the IMC equations – the ‘‘Semi-Analog Monte Carlo’’ (SMC) equations due to Ahrens and Larsen [4] and the Carter– Forest (CF) equations [5] – are prone to similar numerical difficulties, either theoretically or experimentally. This paper provides a new, linear stability analysis technique for generic time discretizations of the nonlinear, grey TRT equations. The technique is developed on the 1-D IMC equations and provides an explicit form of the amplification factor q, which predicts how perturbations about an equilibrium solution are expected to geometrically grow or decay between time steps. Analysis of q is then performed to quantitatively show that (1) the IMC equations are unconditionally stable (jqj < 1) 0021-9991/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2010.11.019 ⇑ Corresponding author. Tel.: +1 505 665 6837; fax: +1 505 667 5921. E-mail addresses: wollaber@lanl.gov (A.B. Wollaber), edlarsen@umich.edu (E.W. Larsen). Journal of Computational Physics 230 (2011) 1528–1546 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp