We develop an asymptotic preserving scheme for the gray radiative transfer equation. Two asymptotic regimes are considered: one is a diffusive regime described by a nonlinear diffusion equation for the material temperature; the other is a free streaming regime with zero opacity. To alleviate the restriction on time step and capture the correct front propagation in the diffusion limit, an implicit treatment is crucial. However, this often involves a large-scale nonlinear iterative solver as the spatial and angular dimensions are coupled. Our idea is to introduce an auxiliary variable that leads to a “redundant” system, which is then solved with a three-stage update: prediction, correction, and projection. The benefit of this approach is that the implicit system is local to each spatial element, independent of angular variable, and thus only requires a scalar Newton's solver. We also introduce a spatial discretization with a compact stencil based on even-odd decomposition. Our method preserves both the nonlinear diffusion limit with correct front propagation speed and the free streaming limit, with a hyperbolic CFL condition.
Bibliographical noteFunding Information:
∗Submitted to the journal’s Computational Methods in Science and Engineering section February 12, 2020; accepted for publication (in revised form) February 28, 2021; published electronically June 16, 2021. https://doi.org/10.1137/20M1318031 Funding: The work of the first author was supported by Science Challenge Project TZ2016002 and by the NSFC through grants NSFC11871340 and NSFC91330203. The work of the second author was partially supported by NSF-DMS 1903420, NSF CAREER-DMS 1846854, and a start-up fund from University of Minnesota. †School of Mathematics, Institute of Natural Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China (firstname.lastname@example.org, email@example.com). ‡School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (wang8818@ umn.edu).
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- Asymptotic preserving
- Front capturing
- Radiative transfer equation