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The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions TRT are challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P$$_N$$ (spherical harmonics) or the S$$_N$$ (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H$$^T_N$$) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P$$_N$$ and S$$_N$$, and indeed reduces to each of these approximations in various limits: H$$^1_N$$ $$\equiv$$ P$$_N$$ and H$$^T_0$$ $$\equiv$$ S$$_T$$. We prove that H$$^T_N$$ results in a system of hyperbolic partial differential equations for all $$T\ge 1$$ and $$N\ge 0$$. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop a numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge-Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H$$^T_N$$ ansatz and the numerical discretizations.
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Stability of structure-aware Taylor methods for tents
Structure-aware Taylor (SAT) methods are a class of timestepping schemes designed for propagating linear hyperbolic solutions within a tent-shaped spacetime region. Tents are useful to design explicit time marching schemes on unstructured advancing fronts with built-in locally variable timestepping for arbitrary spatial and temporal discretization orders. The main result of this paper is that an s s -stage SAT timestepping within a tent is weakly stable under the time step constraint Δ t ≤ C h 1 + 1 / s \Delta t \leq Ch^{1+1/s} , where Δ t \Delta t is the time step size and h h is the spatial mesh size. Improved stability properties are also presented for high-order SAT time discretizations coupled with low-order spatial polynomials. A numerical verification of the sharpness of proven estimates is also included.
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- PAR ID:
- 10433766
- Date Published:
- Journal Name:
- Mathematics of Computation
- Volume:
- 92
- Issue:
- 341
- ISSN:
- 0025-5718
- Page Range / eLocation ID:
- 1061 to 1086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/mcom/3811
