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

非线性动力学方法分析DE数值解的结构

• 批准号: 06650078
• 负责人: HATAUE Itaru
• 金额: $ 1.34万
• 依托单位: Kumamoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06650078/
• 关键词: Numerical Method; Differential Equation; Dynamical System; Computational Fluid Dynamics; 非線型微分方程式; 有限差分法; 数値解析; 流体力学; シミュレーション; カオス理論; フラクタル

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

离散后的有限差分方程解具有复杂的结构。特别是，非线性不稳定性导致离散动力系统的初始条件和边界条件对结构的敏感依赖性。用线性稳定性理论来分析这种复杂的非线性结构是行不通的。在这种情况下，我们需要讨论结构的稳定性和各种摄动的影响。本文将非线性动力学方法应用于简单模型偏微分方程组和实际的CFD计算中。在一维Burgers型方程和反应…两种情况下，由于边值的微小差异，鬼解的定性结构发生变化，而由非线性数值不稳定性引起的解的结构变得简单，对应于空间网格点数的离散动力系统的维度增加更多扩散方程。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。即使我们采用更精确的格式，如高阶Runge-Kutta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。一些类型的误差，如舍入误差、截断误差等，会产生虚假的数值解。然而，目前还没有一个完美的标准来区分伪数值解和真解。另一方面，为了从复杂的数值结果中提取动力学的本质，应用数值渐近线的维度估计也是有效的。特别是通过由计算结果得到的时间序列构造吸引子的关联维数估计，清楚地显示了离散化参数对渐近线结构的依赖性和数值误差的影响，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。结果表明，离散化参数、初始条件和计算方法对流动物理的依赖是非常敏感的，在实际CFD研究中，为了得到物理上合理的数值解，我们还必须注意DT的选择。这意味着，如果计算是稳定的，我们每次都有可能得到其他几个性质不同的稳态解。同时也表明，即使与流动显示结果相差不大，这些非线性动力学也成为提取物理本质的有力武器。从复杂的数值结果中很难表达物理上不同系统之间的定性特征，也很难估计它们之间的定量差异。本文提出的几种方法对计算流体力学领域的研究具有重要意义。较少

项目成果

I.Hataue and H.Takami: "Growth of Three-Dimensional Instability and Onset of Turbulence in Stabilizing ans Destabilizing Two-Stream Mixing Layr" AIAA-Paper. 94-2814. 1-10 (1994)

I.Hataue 和 H.Takami：“稳定和不稳定两流混合层中三维不稳定性的增长和湍流的开始”AIAA 论文。

I.Hataue: "Analysis of Ghost Numerical Solutions of Differential Equation Caused by Nonlinear Instability." AIAA Paper. 1994-0191. 1-9 (1994)

I.Hataue：“非线性不稳定性引起的微分方程幻影数值解的分析”。

I.Hataue: "Application of Nonlinear Dynamics Approaches to the Numerical Study of PDE" AIAA-Paper. 95-0480. 1-8 (1995)

I.Hataue：“非线性动力学方法在偏微分方程数值研究中的应用”AIAA 论文。

I.Hataue and H.Takami: "Growth of Three-Dimensional Instability and Onset of Turbulence in Stabilizing and Destabilizing Two-Stream Mixing Layer in a Mildly" AIAA Paper. 1994-2814. 1-10 (1994)

I.Hataue 和 H.Takami：“在温和的稳定和不稳定两流混合层中三维不稳定性的增长和湍流的开始”AIAA 论文。

I.Hataue: "Analysis of Ghost Numerical Solutions of Differential Equation Caused by Nonlinear Instability" AIAA-Paper. 94-0191. 1-9 (1994)

I.Hataue：“非线性不稳定性引起的微分方程的幽灵数值解的分析”AIAA-论文。