Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion
Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion
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DOI：
10.1006/jcph.2000.6488
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发表时间：
2000-05-20
影响因子：
4.1
通讯作者：
Rider, WJ
中科院分区：
文献类型：
--
作者：
Mousseau, VA;Knoll, DA;Rider, WJ
针对非平衡辐射扩散应用中出现的时变反应扩散系统，提出了一种求解算法。该系统的非线性方程是通过耦合三种数值方法，无雅可比牛顿-克雷洛夫，算子分裂，和多网格线性求解。用不精确牛顿法求解非线性方程组。由于为感兴趣的问题构建雅可比矩阵可能具有挑战性，因此我们采用无雅可比矩阵的牛顿方法实现，其中雅可比矩阵对向量的作用由一阶泰勒级数展开近似。预条件广义最小残差（PGMRES）是用于求解线性系统的克雷洛夫方法，它是由牛顿方法的迭代而来的。该解决方法中的预条件是使用基于物理的分治方法构建的，通常称为算子分裂。这个求解程序用简单的多重网格法对构成前置条件的标量椭圆系统进行了反演。该预条件还解决了具有局部2 x 2块解的方程之间的强耦合，在椭圆解已经解决了细胞间耦合之后应用细胞内耦合。使用此解决方案过程给出的结果证明了它的效率，同时产生最小的内存需求。(C) 2000年学术出版社。
An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical methods, Jacobian-free Newton-Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, we employ a Jacobian-free implementation of Newton's method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov method used to solve the Linear systems that come from the iterations of Newton's method. The preconditioner in this solution method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid methods. The preconditioner also addresses the strong coupling between equations with local 2 x 2 block solves, The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements. (C) 2000 Academic Press.