Efficient Iterative Algorithms for Linear Stability Analysis of Incompressible Flows
Efficient Iterative Algorithms for Linear Stability Analysis of Incompressible Flows
复制标题
不可压缩流线性稳定性分析的高效迭代算法
DOI：
10.1093/imanum/drv003
复制
发表时间：
2015
影响因子：
2.1
通讯作者：
Minghao W. Rostami
中科院分区：
文献类型：
--
作者：
H. Elman;Minghao W. Rostami
动力系统的线性稳定性分析需要找到一系列特征值问题的最右边的特征值。对于大规模系统，已知传统的迭代特征值求解器不能可靠地计算该特征值。Elman&Wu(2013，Lyapunov逆迭代法)提出了一种更稳健的方法来计算大型广义特征值问题的几个最右特征值。暹罗J.矩阵肛门。Appl.，34,1685-1707)和Meerbergen&Spence(2010，纯虚特征值的逆迭代法，并应用于大规模问题的Hopf分支检测。暹罗J.矩阵肛门。应用31,1982-1999)，Lyapunov逆迭代法涉及求解大规模的Lyapunov方程，这又需要求解类似于求解基本偏微分方程组(PDE)的大型稀疏线性系统的解。研究了Lyapunov逆迭代法在不可压缩流动线性稳定性分析中的有效实现方法。通过对Lyapunov方程和基本偏微分方程组的有效求解策略来获得效率。对基于有效预条件方法和基于循环Krylov子空间方法的求解策略进行了测试和比较，并提出了一种改进的Lyapunov求解器，大大节省了计算成本。
Linear stability analysis of a dynamical system entails finding the rightmost eigenvalue for a series of eigenvalue problems. For large-scale systems, it is known that conventional iterative eigenvalue solvers are not reliable for computing this eigenvalue. A more robust method recently developed in Elman & Wu (2013, Lyapunov inverse iteration for computing a few rightmost eigenvalues of large generalized eigenvalue problems. SIAM J. Matrix Anal. Appl., 34, 1685–1707) and Meerbergen & Spence (2010, Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcation in large-scale problems. SIAM J. Matrix Anal. Appl., 31, 1982–1999), Lyapunov inverse iteration, involves solving large-scale Lyapunov equations, which in turn requires the solution of large, sparse linear systems analogous to those arising from solving the underlying partial differential equations (PDEs). This study explores the efficient implementation of Lyapunov inverse iteration when it is used for linear stability analysis of incompressible flows. Efficiencies are obtained from effective solution strategies for the Lyapunov equations and for the underlying PDEs. Solution strategies based on effective preconditioning methods and on recycling Krylov subspace methods are tested and compared, and a modified version of a Lyapunov solver is proposed that achieves significant savings in computational cost.