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

由离散的有限差分方程给出的数值解具有复杂的结构。特别是，非线性不稳定性导致初始条件和边界条件对离散动力系统结构的敏感依赖。对于这种复杂的非线性结构，用线性稳定理论分析是不合适的。在这些情况下，我们需要讨论结构稳定性和各种扰动的影响。在本文件中，本文将非线性动力学方法应用于简单模型偏微分方程和实际CFD计算中，在一维和二维情况下，随着离散动力系统维数（对应于空间网格点数）的增加，鬼解的定性结构随边界值的微小差异而改变，而由非线性数值不稳定性引起的解的结构变得简单。一维Burgers方程和一个反应 ...更多信息 - 扩散方程这些方法对于讨论方程形式和差分格式的依赖性是有效的。即使采用更精确的格式如高阶龙格-库塔格式，鬼解的出现也是不可避免的。伪数值解的出现与格式的精度无关。某些类型的误差，如舍入误差，截断误差等产生虚假的数值解。然而，一个完美的标准，以区分虚假的数值解从真正的解决方案，尚未提供。另一方面，为了从复杂的数值结果中提取动力学的本质，数值渐近线维数估计的应用也是有效的。特别是通过对时间序列构造的吸引子关联维数的估计，揭示了离散参数对渐近线结构的依赖性以及数值误差的影响，并将这些非线性动力学方法应用于实际CFD计算结果的分析。计算结果表明，离散参数、初始条件和计算方法对流动物理性质的依赖性是非常敏感的，因此在应用CFD研究时，为了得到物理上合理的数值解，我们也必须注意离散参数的选择。这意味着如果计算稳定地进行，我们有可能每次得到几个不同性质的稳态解。同时也表明，即使流场显示结果相差不大，这些非线性动力学也是提取物理本质的好武器。从复杂的数值计算结果中很难定性地描述不同物理体系之间的特征，也很难定量地估计不同物理体系之间的差异。本文提出的几种方法是如此有效，为这些目的，他们预计将是重要的研究领域的计算流体力学。少

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