Over the last 10 years there have been major advances in supercomputing, which have triggered adoption of higher and higher resolution in atmospheric numerical models for various scales. And this in turn provides possibility and necessity to use implicit scheme and seek for its efficient algorithms. JFNK (Jacobian-Free Newton-Kry-lov) method, a highly efficient method for solving the implicit nonlinear equation, has been successfully used in a number of fields of computational physics In this paper, the JFNK method and its application in geophysical flows are introduced. The JFNK method has been shown to be more efficient and accurate than the semi-implicit method for a given level of accuracy and a given CPU time. Preliminary results with the advection equation also show that even in the case of strong basic flow, the integration with the implicit scheme keeps well the features of the analytical solution, while apparent distortion occurs with the explicit or semi-implicit schemes. This paper also discusses the applications of the JFNK method in the atmospheric non-hydrostatic model. In a height-based terrain-following coordinate a set of implicit equations for the non-hydrostatic model is derived and the construction of preconditioner using hydrostatic relation and the semi-implicit method is proposed. And the problem solving the 3-D Helmholtz equation, which is commonly involved in non-hydrostatic model, can be reduced to a set of 2-D problems, thus releasing the difficulty of computation. The formulation can not only be used for the finite-difference models but also provide the possibility to use the spectral method in the non-hydrostatic models.
JFNK methodnonlinear equationsimplicit methodpreconditionersemi-implicit methodnonhydro-static model
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