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.