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arXiv 2011.14090math.NAcs.NA
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灰色辐射传输方程的高阶渐近保持间断Galerkin方法

High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations

Tao Xiong, Wenjun Sun, Yi Shi, Peng Song

摘要:

本文将发展一类求解非线性时变灰辐射传输方程的高阶保渐近间断Galerkin方法。受\cite{Peng2020stability}的启发,我们提出在微-宏分解框架下,通过增加一个加权线性扩散项,将非线性GRTE方法进行正则化.在扩散极限下,用双曲型的$Δt=\mathcal{O}(h)$代替抛物线型的$Δt=\mathcal{O}(h^2)$,其中$Δt$和$h$分别为时间步长和网格步长,并且不受光子平均自由程的限制。以将所得到的全局非线性系统从外部迭代循环解耦为具有局部非线性代数方程的线性系。我们的计划被证明是渐近保持和渐近准确的。对一维和二维问题的数值试验表明,我们的格式是高阶的,有效的和高效的。

英文摘要:

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 work \cite{Peng2020stability}, in which stability enhanced high order AP DG methods are proposed for linear transport equations, we propose to pernalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. In the diffusive limit, a hyperbolic, namely $Δt=\mathcal{O}(h)$ where $Δt$ and $h$ are the time step and mesh size respectively, instead of parabolic $Δt=\mathcal{O}(h^2)$ time step restriction is obtained, which is also free from the photon mean free path. The main new ingredient is that 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 from an outer iterative loop. Our scheme is shown to be asymptotic preserving and asymptotically accurate. Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is of high order, effective and efficient.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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