Comprehensive Study on Fundamental and Applied Numerical Algorithms

该项目有一些特定的主题来实现其目标。详细的研究结果将以日语报告的形式发表。主要研究内容如下：1.发展了发展方程的快速大规模算法，并对流体流动问题和浅水问题的有限差分方法和有限元方法进行了分析。阐明了算法和方法的结构。找出了电荷模拟方法中电荷分布的优化算法，并与此相结合，为二维问题的数值分析提供了一种有意义的数值方法。进行了组合结构分析及其应用的算法，其中包括微分-代数方程的算法。分析了线性和非线性方程的迭代解，使其应用于自验证数值计算。在物理现象的数学建模、处理西诺里尼型条件的数值算法或由建模产生的螺旋线分解中，取得了许多结果，常微分方程组和随机微分方程组初值问题的数值算法都是作为离散变量方法发展起来的，它们的稳定性已经得到了很好的阐明。这些研究课题是如此有意义和有前途的，尽管我们的项目取得了很大的成果，但这些研究课题有望在适当的研究项目下继续下去，以取得进一步的成果。

The project has some specified topics to attain its goal. Detailed research results is to be published as a report in Japanese. Summary of several main research topics is as follows :1.Development od fast as well as large-scale algorithms for evolution equations, and analyzes of finite-difference and finite-element methods for fluid flow problems and shallow water problem have been carried out. The structure of the algorithms and methods has been elucidated.An optimization algorithm for the charge distribution in the charge simulation method was found out, and, connected with the above item, a significant numerical method can be achieved for the numerical analysis of 2-dimensional problem.Combinatorial structure analysis has been carried out together with the algorithm of its application, one of which is those for differential-algebraic equations.Iterative solution for linear and nonlinear equations has been analyzed to result in its application to the self-validating numerical computation.Much result has benn attained in the mathematical modelling of physical phenomena, numerical algorithms treating the Signorini-type condition or the spinoidal factorization derived from modelling, and their stability analysis.Numerical algorithms for initial-value problems of ordinary differential equations as well as of stochastic differential equations are developed as discrete variable methods, and their stability has been much elucidated.The research topics are so significant and promising that they are expected to continue under appropriate reseach project for further achievement, although our project attains results in much extent.