Chou Jifan

Computational uncertainty principle in nonlinear ordinary differential equations

In a majority of cases of long-time numerical integration for initial-value problems, round-off error has received little attention. Using twenty-nine numerical methods, the influence of round-off error on numerical solutions is generally studied through a large number of numerical experiments. Here we find that there exists a strong dependence on machine precision (which is a new kind of dependence different from the sensitive dependence on initial conditions), maximally effective computation time (MECT) and optimal stepsize (OS) in solving nonlinear ordinary differential equations (ODEs) in finite machine precision. And an optimal searching method for evaluating MECT and OS under finite machine precision is presented. The relationships between MECT, OS, the order of numerical method and machine precision are found. Numerical results show that round-off error plays a significant role in the above phenomena. Moreover, we find two universal relations which are independent of the types of ODEs, initial values and numerical schemes. Based on the results of numerical experiments, we present a computational uncertainty principle, which is a great challenge to the reliability of long-time numerical integration for nonlinear ODEs.

Existence of the atmosphere attractor

The global asymptotic behavior of solutions for the equations of large scale atmospheric motion with the non-stationary external forcing is studied in the infinite dimensional Hilbert space. Based on the properties of operators of the equations, some energy inequalities and the uniqueness theorem of solutions are obtained. On the assumption that external forces are bounded, the exsitence of the global absorbing set and the atmosphere attractor is proved, and the characteristics of the decay of effect of initial field and the adjustment to the external forcing are revealed. The physical sense of the results is discussed and some ideas about climatic numerical forecast are elucidatedThe global asymptotic behavior of solutions for the equations of large-scale atmospheric motion with the non-stationary external forcing is studied in the infinite-dimensional Hilbert space. Based on the properties of operators of the equations, some energy inequalities and the uniqueness theorem of solutions are obtained. On the assumption that external forces are bounded, the exsitence of the global absorbing set and the atmosphere attractor is proved, and the characteristics of the decay of effect of initial field and the adjustment to the external forcing are revealed. The physical sense of the results is discussed and some ideas about climatic numerical forecast are elucidated.

Predictability of the atmosphere

This paper makes a review on the predictability of the atmosphere. The essential problems of predictability theory, i.e., how a deterministic system changes to an undeterministic system (chaos) and how is the opposite (order within chaos), are discussed. Some applications of predictability theory are given.

Strategy and methodology of dynamical analogue prediction

In order to effectively improve numerical prediction level by using current models and data, the strategy and methodology of dynamical analogue prediction (DAP) is deeply studied in the present paper. A new idea to predict the prediction errors of dynamical model on the basis of historical analogue information is put forward so as to transform the dynamical prediction problem into the estimation problem of prediction errors. In terms of such an idea, a new prediction method of final analogue correction of errors (FACE) is developed. Furthermore, the FACE is applied to extra-seasonal prediction experiments on an operational atmosphere-ocean coupled general circulation model. Prediction results of summer mean circulation and total precipitation show that the FACE can to some extent reduce prediction errors, recover prediction variances, and improve prediction skills. Besides, sensitive experiments also show that predictions based on the FACE are evidently influenced by the number of analogues, analogue-selected variables and analogy metric.

A new difference scheme with multi-time levels

In view of making the best use of information coming from past observational data, a new difference scheme with multi-time levels (p>3) is suggested. Some mathematical characteristics of the scheme, which is called the retrospective scheme, are discussed. The numerical results of some examples show that the calculation accuracy of linear and nonlinear advection equations computed with the retrospective scheme is higher than that obtained via the leapfrog scheme. The scheme can be applied to many fields, such as meteorology, engineering, physics, astronautics, environment and economy etc, where systematic observations are made normally.

Three-dimensional decomposition method of global atmospheric circulation

By adopting the idea of three-dimensional Walker, Hadley and Rossby stream functions, the global atmospheric circulation can be considered as the sum of three stream functions from a global perspective. Therefore, a mathematical model of three-dimensional decomposition of global atmospheric circulation is proposed and the existence and uniqueness of the model are proved. Besides, the model includes a numerical method leading to no truncation error in the discrete three-dimensional grid points. Results also show that the three-dimensional stream functions exist and are unique for a given velocity field. The mathematical model shows the generalized form of three-dimensional stream functions equal to the velocity field in representing the features of atmospheric motion. Besides, the vertical velocity calculated through the model can represent the main characteristics of the vertical motion. In sum, the three-dimensional decomposition of atmospheric circulation is convenient for the further investigation of the features of global atmospheric motions.

Analogue correction method of errors and its application to numerical weather prediction

In this paper, an analogue correction method of errors (ACE) based on a complicated atmospheric model is further developed and applied to numerical weather prediction (NWP). The analysis shows that the ACE can effectively reduce model errors by combining the statistical analogue method with the dynamical model together in order that the information of plenty of historical data is utilized in the current complicated NWP model. Furthermore, in the ACE, the differences of the similarities between different historical analogues and the current initial state are considered as the weights for estimating model errors. The results of daily, decad and monthly prediction experiments on a complicated T63 atmospheric model show that the performance of the ACE by correcting model errors based on the estimation of the errors of 4 historical analogue predictions is not only better than that of the scheme of only introducing the correction of the errors of every single analogue prediction, but is also better than that of the T63 model.

ON THE PREDICTABILITY OF CHAOTIC SYSTEMS WITH RESPECT TO MAXIMALLY EFFECTIVE COMPUTATION TIME

The round-off error introduces uncertainty in the numerical solution. A computational uncertainty principle is explained and validated by using chaotic systems, such as the climatic model, the Rossler and super chaos system. Maximally effective computation time (MECT) and optimal stepsize (OS) are discussed and obtained via an optimal searching method. Under OS in solving nonlinear ordinary differential equations, the self-memorization equations of chaotic systems are set up, thus a new approach to numerical weather forecast is described.

Retrospective time integral scheme and its applications to the advection equation

To put more information into a difference scheme of a differential equation for making an accurate prediction, a new kind of time integration scheme, known as the retrospective (RT) scheme, is proposed on the basis of the memorial dynamics. Stability criteria of the scheme for an advection equation in certain conditions are derived mathematically. The computations for the advection equation have been conducted with its RT scheme. It is shown that the accuracy of the scheme is much higher than that of the leapfrog (LF) difference scheme.

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

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.