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首页> 外文期刊>SIAM Journal on Numerical Analysis >ON PRECONDITIONED ITERATIVE METHODS FOR CERTAIN TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS
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ON PRECONDITIONED ITERATIVE METHODS FOR CERTAIN TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS

机译：时滞偏微分方程的预处理迭代方法研究

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摘要

When the Newton method or the fixed-point method is employed to solve the systems of nonlinear equations arising in the sinc-Galerkin discretization of certain time-dependent partial differential equations, in each iteration step we need to solve a structured subsystem of linear equations iteratively by, for example, a Krylov subspace method such as the preconditioned GMRES. In this paper, based on the tensor and the Toeplitz structures of the linear subsystems we construct structured preconditioners for their coefficient matrices and estimate the eigenvalue bounds of the preconditioned matrices under certain assumptions. Numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods. It has been shown that a combination of the Newton/fixed-point iteration with the preconditioned GMRES method is efficient and robust for solving the systems of nonlinear equations arising from the sinc-Galerkin discretization of the time-dependent partial differential equations.
机译：当采用牛顿法或定点法求解某些时变偏微分方程在sinc-Galerkin离散化中产生的非线性方程组时，在每个迭代步骤中，我们都需要迭代求解结构化的线性方程组子系统通过例如Krylov子空间方法，例如预处理的GMRES。在本文中，基于线性子系统的张量和Toeplitz结构，我们为它们的系数矩阵构造了结构化的预处理器，并在某些假设下估计了预处理条件的特征值范围。数值例子说明了所提出的预处理方法的有效性。已经表明，牛顿/定点迭代与预处理GMRES方法的组合对于求解由时变偏微分方程的sinc-Galerkin离散化引起的非线性方程组是有效且鲁棒的。

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