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

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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