Computational uncertainty principle in nonlinear ordinary differential equations——Ⅱ.Theoretical analysis

The error propagation for general numerical method in ordinarydifferential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components of round-off error occurring in floating-point computation are fully detailed. By introducing a new kind of recurrent inequality, the classical error bounds for linear multistep methods are essentially improved, and joining probabilistic theory the “normal” growth of accumulated round-off error is derived. Moreover, a unified estimate for the total error of general method is given. On the basis of these results, we rationally interpret the various phenomena found in the numerical experiments in part I of this paper and derive two universal relations which are independent of types of ODEs, initial values and numerical schemes and are consistent with the numerical results. Furthermore, we give the explicitly mathematical expression of the computational uncertainty principle and expound the intrinsic relation between two uncertainties which result from the inaccuracies of numerical method and calculating machine.

Recent Advances in Predictability Studies in China (1999-2002)

Since the last International Union of Geodesy and Geophysics (IUGG) General Assembly (1999), the predictability studies in China have made further progress during the period of 1999-2002. Firstly, three predictability sub-problems in numerical weather and climate prediction are classified, which are concerned with the maximum predictability time, the maximum prediction error, and the maximum allowable initial error, and then they are reduced into three nonlinear optimization problems. Secondly, the concepts of the nonlinear singular vector (NSV) and conditional nonlinear optimal perturbation (CNOP) are proposed, which have been utilized to study the predictability of numerical weather and climate prediction. The results suggest that the nonlinear characteristics of the motions of atmosphere and oceans can be revealed by NSV and CNOP. Thirdly, attention has also been paid to the relations between the predictability and spatial-temporal scale, and between the modei predictability and the machine precision, of which the investigations disclose the importance of the spatial-temporal scale and machine precision in the study of predictability. Also the cell-to-cell mapping is adopted to analyze globally the predictability of climate, which could provide a new subject to the research workers. Furthermore, the predictability of the summer rainfall in China is investigated by using the method of correlation coefficients. The results demonstrate that the predictability of summer rainfall is different in different areas of China. Analysis of variance, which is one of the statistical methods applicable to the study of predictability, is also used to study the potential predictability of monthly mean temperature in China, of which the conclusion is that the monthly mean temperature over China is potentially predictable at a statistical significance Ievel of 0.10. In addition, in the analysis of the predictability of the T106 objective analysis/forecasting field, the variance and the correlation coemcient are calculated to explore the distribution characteristics of the mean-square errors. Finally, the predictability of short-term climate prediction is investigated by using statistical methods or numerical simulation methods. It is demonstrated that the predictability of short-terrn climate in China depends not only on the region of China being investigated, but also on the time scale and the atmospheric internai dynamical process.

Interval of effective time-step size for the numerical computation of nonlinear ordinary differential equations

The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations（ODEs） should use appropriately sized time steps to obtain reliable solutions.However,the interval of effective step size（IES） has not been thoroughly explored theoretically.In this paper,by using a general estimation for the total error of the numerical solutions of ODEs,a method is proposed for determining an approximate IES by translating the functions for truncation and rounding errors.It also illustrates this process with an example.Moreover,the relationship between the IES and its approximation is found,and the relative error of the approximation with respect to the IES is given.In addition,variation in the IES with increasing integration time is studied,which can provide an explanation for the observed numerical results.The findings contribute to computational step-size choice for reliable numerical solutions.

Computational Fluid Dynamics Uncertainty Analysis for Simulations of Roll Motions for a 3D Ship

The roll motions are influenced by significant viscous effects such as the flow separation.The 3D simulations of free decay roll motions for the ship model DTMB 5512 are carried out by Reynold averaged NavierStokes(RANS) method based on the dynamic mesh technique.A new moving mesh technique is adopted and discussed in details for the present simulations.The purpose of the research is to obtain accurate numerical prediction for roll motions with their respective numerical/modeling errors and uncertainties.Errors and uncertainties are estimated by performing the modern verification and validation(V&V) procedures.Simulation results for the free-floating surface combatant are used to calculate the linear,nonlinear damping coefficients and resonant frequencies including a wide range of forward speed.The present work can provide a useful reference to calculate roll damping by computational fluid dynamics(CFD) method and simulate a general ship motions in waves.