Studies on the treatment of Numerical Calculations Including Considerable Errors for Construction of Proper Mathematical Discrete models

在这项研究中，我们要分析的结构动力系统，这是由离散的非线性微分方程（DE）从一些观点，如分析和/或数值方法，定性和/或定量的。数值非线性动力学方法，如渐近数值解。研究了用隐式格式计算的渐近数值解的结构。对Burgers方程的全隐格式及其几种线性化格式进行了分析讨论和数值试验。此外，我们还从四阶人工粘性项的影响出发，研究了不可压缩流体方程鬼数值解的特征。其次，我们尝试从四阶人工粘性效应的角度来研究不可压缩流体方程鬼数值解的特性 ...更多信息 第二，通过计算近似广义维数，特别是关联维数，来估计非线性微分方程离散化后的数值解的维数。计算了由直接求解Navier-Stokes方程得到的数值结果中的一个变量的时间序列所构造的吸引子的维数。第三，本文的研究目的是应用IPNS（Infinite Precision Numerical Simulation）方法消除数值误差。求解了椭圆算子的柯西问题和第一类积分方程。另一个应用是自由边界问题中吸引子的数值计算。少

In this study we want to analyze the structure of dynamical systems which are produced by discretizing the nonlinear differential equations(DEs) from some viewpoints such as analytical and/or numerical approaches, and qualitative and/or quantitative ones. The numerical nonlinear dynamics approaches such as asymptotic numerical solutions. The structure of the asymptotic numerical solutions which were calculated by using implicit schemes were studied. Analytical discussions and numerical tests for fully implicit schemes of the Burgers' equation and several types of their linearized schemes were done. Furthermore, we tried to investigate the characteristics of ghost numerical solutions of incompressible fluid equations from viewpooints of the effect of popular fourth order artificial viscosity terms. Next, we tried to investigate the characteristics of ghost numerical solutions of incompressible fluid equations from the viewpoints of the effect of popular fourth order artificial viscosity … More terms and to discuss influence of the condition for convergence in iterative steps in solving the linear systems on the structure of numerical solutions.Second purpose of the present study is try to evaluate the dimensions of numerical solutions which are produced by discretizing the noonlinear differential equations by calculating the approximately generalized dimension, especially the correlation dimension. The dimensions of attractors constructed from the time series of one of the valiables in the numerical results which were given by solving Navier-Stokes equations directly are calculated. Futhermore, wavelet analysis is applied in order to clarify the difference of the complicated structure of numerical solutions in detail.Third purpose of the present study is that we apply IPNS(Infinite Precision Numerical Simulation) approach in order to remove numerical errors. A Cauchy problem of an elliptic operator and an integral equation of the first kind are solved. Another application is numerical computation of attractors in free boundary problems. Less