An adaptive nested source term iteration for radiative transfer equations
An adaptive nested source term iteration for radiative transfer equations
复制标题
辐射传递方程的自适应嵌套源项迭代
DOI：
10.1090/mcom/3505
复制
发表时间：
2020
影响因子：
2
通讯作者：
Mula, Olga
中科院分区：
文献类型：
--
作者：
Dahmen, Wolfgang;Gruber, Felix;Mula, Olga
本文提出了一种新的辐射传输方程数值解方法，并证明了该方法的后验误差界。稳定的参数传输方程和相应的辐射传输方程的彼得罗夫-伽辽金型变分公式发挥了关键作用。这使我们能够在一个合适的，无限维的函数空间，保证收敛与固定的误差减少每一步制定一个迭代。然后，数值方案基于在动态更新的精度容限内近似实现该迭代，该精度容限仍然确保收敛到精确解。为了推进此迭代，需要在适当收紧的精度公差内执行两个操作。首先，需要在与当前精度水平相当的容差内将全局散射算子近似地应用于当前的反射率。第二，需要求解参数相关的线性输运方程，再次以迭代所需的精度。为了确保阶段相关的误差容限得到满足，必须采用严格的后验误差界，在我们的情况下，休息的不连续彼得罗夫-伽辽金（DPG）计划。这些后验界不仅对保证扰动迭代的收敛性至关重要，而且还用于生成适应参数依赖的空间网格。这将大大降低整体计算复杂度。由于只应用全局算子，我们避免了求解具有密集矩阵的线性系统的需要。此外，通过低秩近似和矩阵压缩技术，加速了全局散射体的近似应用。理论研究结果的说明和补充的数值实验与非平凡的散射内核。引用
We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds for thenorm. A key role is played by stable Petrov–Galerkin-type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite-dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov–Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer is accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels. References
登录
查看更多内容
静态辐射传输的 Lp 理论
DOI：
--
发表时间：
2013
期刊：
影响因子：
--
作者：
H. Egger;M. Schlottbom
通讯作者：
M. Schlottbom
用于求解不连续 Petrov-Galerkin 有限元偏微分方程的 DUNE-DPG 库
DOI：
10.11588/ans.2017.1.27719
发表时间：
2016
期刊：
arXiv: Numerical Analysis
影响因子：
--
作者：
Felix Gruber;Angela Klewinghaus;Olga Mula
通讯作者：
Olga Mula
影响因子：
2
作者：
ALPERT, BK
通讯作者：
ALPERT, BK
$L_2$-三维中子输运离散坐标法的误差估计
DOI：
--
发表时间：
1988
期刊：
影响因子：
--
作者：
M. Asadzadeh
通讯作者：
M. Asadzadeh
DOI：
--
发表时间：
2019
期刊：
影响因子：
--
作者：
Felix Gruber;W. Dahmen;M. Torrilhon
通讯作者：
M. Torrilhon