Time-step limits for a Monte Carlo Compton-scattering method

Compton scattering is an important aspect of radiative transfer in high energy density applications. In this process, the frequency and direction of a photon are altered by colliding with a free electron. The change in frequency of a scattered photon results in an energy exchange between the photon and target electron and energy coupling between radiation and matter. Canfield, Howard, and Liang have presented a Monte Carlo method for simulating Compton scattering that models the photon-electron collision kinematics exactly. However, implementing their technique in multiphysics problems that include the effects of radiation-matter energy coupling typically requires evaluating the material temperature at its beginning-of-time-step value. This explicit evaluation can lead to unstable and oscillatory solutions. In this paper, we perform a stability analysis of this Monte Carlo method and present time-step limits that avoid instabilities and nonphysical oscillations by considering a spatially independent, purely scattering radiative-transfer problem. Examining a simplified problem is justified because it isolates the effects of Compton scattering, and existing Monte Carlo techniques can robustly model other physics (such as absorption, emission, sources, and photon streaming). Our analysis begins by simplifying the equations that are solved via Monte Carlo within each time step using the Fokker-Planck approximation. Next, we linearize these approximate equations about an equilibrium solution such that the resulting linearized equations describe perturbations about this equilibrium. We then solve these linearized equations over a time step and determine the corresponding eigenvalues, quantities that can predict the behavior of solutions generated by a Monte Carlo simulation as a function of time-step size and other physical parameters. With these results, we develop our time-step limits. This approach is similar to our recent investigation of time discretizations for the Compton-scattering Fokker-Planck equation.