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

在这项研究中，我们想从分析和/或数值方法，以及定性和/或定量的观点出发，分析非线性微分方程组(DES)离散化所产生的动力系统的结构。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型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，我们还试图从流行的四阶人工粘性项的影响的角度来研究不可压缩流体方程的鬼影数值解的特征。接下来，我们尝试从流行的四阶人工粘性…的影响的角度来研究不可压缩流体方程的重影数值解的特征。本文的第二个目的是通过计算近似广义维，特别是关联维来计算离散的非线性微分方程组所产生的数值解的维数。计算了由直接求解N-S方程得到的数值结果中的一个变量的时间序列构成的吸引子的维度。此外，为了更详细地阐明数值解复杂结构的差异，本文还应用了小波分析。第三个目的是应用无限精度数值模拟(IPNS)方法来消除数值误差。求解了一类椭圆算子的柯西问题和一类第一类积分方程解。另一个应用是自由边界问题中吸引子的数值计算。较少

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