Precision Improvement of Numerical Simulation Including Iteration Processes for Nonlinear Evolution Equations

为了构造极大地消除误差影响的高贵格式，在求解线性方程组的迭代过程中，我们考虑了数值解收敛的合理条件的设置和收敛速度的提高。此外，我们将它们应用于实际的流体计算问题和交通拥堵问题，以分析控制非线性现象的几个因素。首先，我们从数值实验、吸引子非线性结构的可视化和统计考虑等几个角度分析了非线性微分方程离散化所产生的动力系统的结构。具体地说，阐明了离散动力系统数值解的基本结构，提出了判定解可靠性的判据，并从概率论的角度讨论了随机形式的误差插入对随机微分方程数值解结构的影响。具体地说，我们试图研究数值解的结构对随机误差插入的依赖性。作为基础性研究，考虑了基于确定性Logistic方程的随机微分方程，讨论了噪声大小与数值解特征之间的关系。在此基础上，讨论了不可压缩流体方程数值解的结构对求解联立方程时随机误差插入的依赖关系。其次，利用无限精度数值模拟软件IPNS对第一类积分方程解进行了直接数值模拟。数值结果的精度令人满意。此外，它们还显示了一些有趣的事实。这些数值结果表明，IPNS为此类反问题的数值分析提供了便利。

For the purpose of construction of noble schemes in which influence of errors are extremely removed, we did consideration about setting of the reasonable condition for convergence of numerical solutions in iterative steps in solving the linear systems and about speed-up of convergence. In addition, we applied them to the issues of real fluid calculation and the problem of traffic jam in order to analyze several factors which govern nonlinear phenomena. First, we analyzed the structure of dynamical systems which are produced by discretizing the nonlinear differential equations from several viewpoints such as numerical experiments, visualization of nonlinear structure of attractors and statistical consideration. Concretely, we clarified essential structure of numerical solutions in discrete dynamical system and proposed the criterion to judge reliability of solutions.Next, we tried to discuss how structure of numerical solutions of a stochastic differential equation changes by the insertion of errors with the random style from the viewpoints of probabilistic approaches. Concretely, we tried to investigate the dependence of the structure of numerical solutions on insertion of random errors. As a fundamental study, the stochastic differential equation based on the deterministic logistic differential equation was considered and the relation between the size of noise and characteristics of obtained numerical solutions was discussed. Furthermore, we tried to discuss the dependence of the structure of numerical solutions of incompressible fluid equations on insertion of random errors in solving simultaneous equations.Next, direct numerical simulation to an integral equation of the first kind was carried out by using IPNS(Infinite-Precision Numerical Simulation). Numerical results are very satisfactory in accuracy. Moreover, they also show some interesting facts. These numerical results show IPNS facilitates numerical analysis for such inverse problems.