High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations. (English) Zbl 07536794
Summary: In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the works in [S. Boscarino et al., SIAM J. Sci. Comput. 36, No. 2, 377–395 (2014; Zbl 1426.76455); Z. Peng et al., J. Comput. Phys. 415, Article ID 109485, 35 p. (2020; Zbl 1440.65147)], we propose to penalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. A hyperbolic, namely \(\Delta t = \mathcal{O}(h)\) in the transport regime where \(\Delta t\) and \(h\) are the time step and mesh size respectively, and unconditional stability instead of parabolic time step restriction \(\Delta t = \mathcal{O}(h^2)\) in the diffusive regime are obtained, which are also free from the photon mean free path. We further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations within each outer iterative loop. For the resulting scheme, only an implicit system for the macroscopic variable needs to be solved, while the microscopic variable can be updated explicitly. Besides, the nonlinear implicit system is decoupled to a linear positive definite system followed by nonlinear algebraic equations due to the Picard iteration. Namely, high dimension, implicit treatment and nonlinearity are all decoupled. Our scheme is shown to be AP and asymptotically accurate (AA). Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is high order accurate, effective and efficient.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 82Cxx | Time-dependent statistical mechanics (dynamic and nonequilibrium) |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
Keywords:
gray radiative transfer equations; asymptotic preserving; discontinuous Galerkin method; high order; micro-macro decompositionReferences:
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