具有各向同性散射的尺度辐射传递方程的球谐不连续伽辽金方法的数值分析

IF 2.3 2区 数学 Q1 MATHEMATICS, APPLIED
Qiwei Sheng, Cory D Hauck, Yulong Xing
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引用次数: 0

摘要

在平均自由程$\varepsilon $趋于零的高扩散状态下,辐射传递方程具有由扩散方程和相应的边界条件控制的渐近特性。通常,解决该问题的数值方案具有截断误差,其中包含$\varepsilon ^{-1}$贡献,导致小$\varepsilon $的非均匀收敛。这种现象需要高分辨率的离散化,而这降低了数值格式在扩散极限下的性能。在本文中,我们首先对标度球谐($P_{N}$)辐射传递方程提供了一个先验估计。然后,对标度辐射传递方程的球谐不连续伽辽金(DG)方法进行了误差分析,结果表明,在一些附加假设下,其解在$\varepsilon $内一致收敛于标度辐射传递方程的解。进一步给出了在直角网格上考虑迎风通量的DG方法的最优收敛结果。当使用最多$k$次的张量积多项式时,得到$\left (1+\mathcal{O}(\varepsilon )\right )h^{k+1}$(其中$h$是最大元素长度)的误差估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering
In highly diffusion regimes when the mean free path $\varepsilon $ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon ^{-1}$ contribution that leads to a nonuniform convergence for small $\varepsilon $. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a priori estimates for the scaled spherical harmonic ($P_{N}$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some additional assumptions, its solutions converge uniformly in $\varepsilon $ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left (1+\mathcal{O}(\varepsilon )\right )h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.
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来源期刊
IMA Journal of Numerical Analysis
IMA Journal of Numerical Analysis 数学-应用数学
CiteScore
5.30
自引率
4.80%
发文量
79
审稿时长
6-12 weeks
期刊介绍: The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.
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