Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem
Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem
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DOI：
10.1016/j.jcp.2014.06.028
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发表时间：
2014-10-01
影响因子：
4.1
通讯作者：
Planas, Ramon
中科院分区：
文献类型：
--
作者：
Badia, Santiago;Martin, Alberto F.;Planas, Ramon
热耦合不可压缩无感磁流体动力学 (MHD) 问题通过热耦合模拟带电流体在外部电磁场影响下的流动。对于感兴趣的实际情况，该偏微分方程组是强耦合和高度非线性的。因此，为了捕获当前问题的不同物理尺度，完全隐式的时间积分方案是非常可取的。然而，求解由此类算法产生的多物理场线性方程组是一项非常具有挑战性的任务，需要高效且可扩展的预处理器。在这项工作中，设计并测试了一系列新的递归块 LU 预处理器，用于求解热耦合无感 MHD 方程。这些预处理器是在将全耦合矩阵分解为每个变量（速度、压力、电流密度、电势和温度）的单物理问题后获得的，这些变量可以通过例如使用预处理域分解算法得到最佳解决。主要思想是将原始矩阵排列成（任意）2 x 2 块矩阵，并考虑通过近似相应的 Schur 补集获得的 LU 预处理器。对于 LU 预处理器中的每一个对角块，如果它涉及不止一种类型的未知数，我们就会以递归方式进行相同的处理。这种方法以抽象的方式表述，可以直接应用于其他多物理场问题。此外，我们精确地解释了此类预处理器的代码实现的灵活且通用的软件设计。 (C) 2014 Elsevier Inc. 保留所有权利。
The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 x 2 block matrix, and consider an LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknowns, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners. (C) 2014 Elsevier Inc. All rights reserved.