Numerical Analysis of a Hybrid Method for Radiation Transport

In this work, we prove rigorous error estimates for a hybrid method introduced in [Hauck, Cory D, and Ryan G McClarren. 2013. Multiscale Modeling & Simulation 11 (4):1197–1227] for solving the time-dependent radiation transport equation (RTE). The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domain, with constant material cross-sections of arbitrary size. To focus the analysis, we assume the high-resolution method for the uncollided equation is, in fact, an exact solution and the collided part is approximated in angle via a spherical harmonic expansion ((Formula presented.) method). Using a nonstandard set of semi-norms, we obtain estimates of the form (Formula presented.) where (Formula presented.) denotes the regularity of the solution in angle, (Formula presented.) and (Formula presented.) are scattering parameters, (Formula presented.) is the time-step before reinitialization, and C is a complicated function of (Formula presented.) (Formula presented.) and (Formula presented.) These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic (Formula presented.) method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes.