Implicit Asymptotic Preserving Method for Linear Transport Equations
Implicit Asymptotic Preserving Method for Linear Transport Equations
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DOI：
10.4208/cicp.oa-2016-0105
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发表时间：
2017-07-01
影响因子：
3.7
通讯作者：
Wang, Li
中科院分区：
文献类型：
--
作者：
Li, Qin;Wang, Li
辐射传递方程的计算是昂贵的，主要是由于两个刚性项：运输项和碰撞算子。前者的刚性来自于粒子（如光子）以光速传播的事实，而后者则是由于光学厚区域的强散射。我们研究了该方程的全隐格式来考虑刚度。隐式处理中的主要挑战是空间坐标和角坐标之间的耦合，这需要大尺寸的待求逆矩阵，该矩阵也是病态的并且不一定对称。我们的主要思想是利用病态矩阵的谱结构来构造一个预条件子，该预条件子沿着空间和角度依赖性的精细分裂，显著地改善了条件数，并允许无矩阵处理。我们还设计了一个快速求解器来计算这个预条件提前明确。我们的方法被证明是有效的扩散和自由流限制，计算成本是最先进的方法。各种例子，包括各向异性散射和二维问题，以验证我们的方法的有效性。
The computation of the radiative transfer equation is expensive mainly due to two stiff terms: the transport term and the collision operator. The stiffness in the former comes from the fact that particles (such as photons) travel at the speed of light, while that in the latter is due to the strong scattering in the optically thick region. We study the fully implicit scheme for this equation to account for the stiffness. The main challenge in the implicit treatment is the coupling between the spacial and angular coordinates that requires the large size of the to-be-inverted matrix, which is also ill-conditioned and not necessarily symmetric. Our main idea is to utilize the spectral structure of the ill-conditioned matrix to construct a pre-conditioner, which, along with an exquisite split of the spatial and angular dependence, significantly improve the condition number and allows a matrix-free treatment. We also design a fast solver to compute this pre-conditioner explicitly in advance. Our method is shown to be efficient in both diffusive and free streaming limit, and the computational cost is comparable to the state-of-the-art method. Various examples including anisotropic scattering and two-dimensional problems are provided to validate the effectiveness of our method.