Uncertainty of the numerical solution of a nonlinear system’s long-term behavior and global convergence of the numerical pattern

Abstract

The computational uncertainty principle in nonlinear ordinary differential equations makes the numerical solution of the long-term behavior of nonlinear atmospheric equations have no meaning. The main reason is that, in the error analysis theory of present-day computational mathematics, the non-linear process between truncation error and rounding error is treated as a linear operation. In this paper, based on the operator equations of large-scale atmospheric movement, the above limitation is overcome by using the notion of cell mapping. Through studying the global asymptotic characteristics of the numerical pattern of the large-scale atmospheric equations, the definitions of the global convergence and an appropriate discrete algorithm of the numerical pattern are put forward. Three determinant theorems about the global convergence of the numerical pattern are presented, which provide the theoretical basis for constructing the globally convergent numerical pattern. Further, it is pointed out that only a globally convergent numerical pattern can improve the veracity of climatic prediction.