预处理JFNK方法及其在大气方程中的初步应用

为提高数值求解大气方程的速度, 研究了预处理JFNK（Jacobian-Free Newton-Krylov）方法及其在大气方程中的应用。这是一种非线性外循环Newton迭代与线性内循环Krylobv迭代相结合的快速算法，其优点是进行外循环Newton迭代时不要求Jacobian矩阵的形成和存储；它的有效性取决于内循环中线性系统的预处理。首先介绍了JFNK算法，然后以浅水波方程为例，描述了非线性残值的形成、预处理矩阵的构造及其在JFNK算法中的应用。试验结果表明：对内循环线性系统进行适当的预处理，能大幅度提高JFNK算法的运算速度。因而，JFNK是一种值得在大气方程中推广应用的方法。

The Preconditioning JFNK Method and Its Initial Application in Weather Equations

The preconditioning JFNK (Jacobian-Free Newton-Krylov) method and its application in weather equations are studied to improve the computation speed in solving weather equations numerically. This method is based on the fully implicit discrete difference equations. It is a fast algorithm by the combination of the Newton iteration and the Krylov iteration. Its advantage is that the formation and storage of the Jacobian matrix in the outer Newton iteration is avoided. Its effectiveness depends on the preconditioning method to the Krylov inner loop. Firstly, the JFNK algorithm is introduced, and then an example about shallow water equations was given. The formation of nonlinear residuals, the construction of preconditioning matrix, and its application to JFNK are described in the example. It is shown that the computation speed of the JFNK algorithm can be greatly improved by a proper preconditioning for the linear system in the inner loop. Therefore, the algorithm is potential in the application of weather equations.