计算地球物理流体力学中的非线性不稳定问题(续)

四、算子差分方程和谱方法的非线性稳定问题§10 计算稳定性和能量守恒性与算子非负性的关系本节引进算子方程,并推广§6和§8的一些结果。对于非定常流体力学方程,往往都可以化为如下形式的“发展方程”:其中F=F(x,t)是待求函数,是一个非线性算子,X=X(x_1,…,x_k)是空间坐标,k是空间维数,t是时间坐标。例如,对于正压原始方程

ON THE NONLINEAR COMPUTATIONAL INSTABILITY IN COMPUTATIONAL GEOPHYSICAL FLUID DYNAMICS

In past few years, several problems about nonlinear computational stability have been studied and discussed for the typical one-dimensional nonlinearadvcction equation the two-d imcnsional vorticity equation and the nonlinear evolution equation . Inthis paper, the following points arc discussed:1. More nonlinear computational ins table examples of Richlmycr-typc and of Fornberg-type are given.2. The relations between the computational stability and the conservation property of energy are examined. 3. It is shown that nonlinear computational instability may occur even for "I illy scheme" and "Arakawa scheme" of instantaneous energy conservation which are widely adopted in numerical weather prediction.4. The problems of computational stability of those equations which arise in the Galcrkin finite clement method and the spectral method arc discussed.5. The effects of dissipation on computational stability arc analysed.6. The generation mechanism of nonlinear computational instability'is discussed. Particularly, the effects of conservation property of energy and of initial value condition on the computational stability arc emphasized.7. Some methods for construction of the scheme of energy conservation arc-summarized.8. The scheme of energy conservation is applied to the calculation of sea currents and satisfactory results of computation are obtained.