Analysis of the Structure of Numerical Solution of DE by Nonlinear Dynamics Approaches

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Numerical solutions which are given from the discretized finite difference equation have complicated structures. Especially, nonlinear instability causes the sensitive dependence of the initial condition and boundary conditions on the structure of the discrete dynamical system. To analyze such complicated nonlinear structure from the viewpoint of linear stability theory is not effective. In these cases, we need the discussion about the structural stability and the effects of various perturbations. In the present paper, we applied the nonlinear dynamics approaches to the simple model PDE cases and practical CFD computation.The qualitative structure of ghost solutions changes by the small difference of the boundary values and the structure of solutions caused by the nonlinear numerical instabilities becomes simple as the dimension of discrete dynamical system which corresponds to the number of spatial grid points increases in both cases of one-dimensional Burgers' equation and a reaction … More -diffusion equation. These approaches are effective to discuss the depencdence of the equation form and difference schemes. Even if we adopt the more accurate scheme such as the higher-order Runge-Kutta scheme, the appearance of ghost solutions may be inevitable. The appearance of spurious numerical solutions is irrelevant to the accuracy of the scheme. Some types of errors such as rounding error, truncation error and so on produce spurious numerical solutions. However, a perfect criterion to distinguish the spurious numerical solutions from the true solutions is not yet available. On the other hand, application of the estimation of dimension of numerical asymptotes is also effective in order to extraxct the essence of dynamics fron the complicated numerical results. Especially, the dependence of the discretized parameter on the structure of the asymptotes and the influence of numerical errors were clearly shown by the estimation of correlation dimension of attractors constructed by using the time series from computational results.Application of these nonlinear dynamics approaches to the analyzes of the results of practical CFD computations was also done. It was shown that the dependences of the discretized parameter, initial condition and computationl technique on the physics of flow are so sensitive that we also have to pay much attention to the selection of Dt in the case of ptactical CFD stusied in order to get the physically reasonable numerical solutions. This means that there is a possibility for us to get several other qualitatively different steady atate solutions everytime if the calculations were performed stablely. Simultaneously, it is shown that these nonlinear dynamics become the good weapon in order to extract the essence of physics even if few difference can be seen from the flow visualization results. It is difficult to express the qualitaive feature and to estimate the quantitative difference between the physically different system from the complicated numerical results. The several methods proposed in the present paper are so effective for such purposes that they are expected to be important to studies in the field of computational fluid dynamics. Less