Numerical Analysis of Evolution Systems

该项目已经执行。旨在从解析和数值两个角度研究演化系统。已知该系统出现在科学和工程的广泛领域的数学建模中。所列出的研究者以及他们的合作者所得到的结果如下：时滞或随机系统的离散变量方法。对其稳定性进行了线性和非线性分析。物理应用也被认为是。长期积分的哈密尔顿系统。揭示了汉密尔顿系统与其辛数值积分器之间的代数对应关系。讨论了其保性和辛性。微分方程的自验证数值解。成功地实现了抛物型偏微分方程的求解。常微分方程的情况也得到了解决。演化系统的进化论。通过对收敛性、稳定性和高性能数值方法的分析，研究了跨步并行和跨方法并行。为了达到这一目的，对Runge-Kutta格式作了各种修改。本文的研究成果已经或将要发表在多篇文章中，其中一部分已经得到了验证。下一页列出了其中的一些。

The project. has been carried out., aiming to study the evolution systems in both analytical and numerical points of view. The systems are known to appear in mathematical modelling of broad area in science and engineering. The results obtained by the listed investigators as well as by their collaborators are as follows.Discrete variable methods for systems with time-delay or randomness. Linear and nonlinear analysis has been done for their stability. Physical applications are also considered.Long-term integration of Hamiltonian systems. Algebraic correspondence between Hamilton systems and their symplectic numerical integrators is unveiled. Symplecticness and the conservative quantities are also studied.Self-validating numerical solution of differential equations. The solution of parabolic partial differential equations is successfully achieved. Ordinary differential equation case is also tackled.Parallelism for evolution systems. Across-the-step and across-the-method parallelism have been studied with analysis of convergence.Stable and high-performance numerical methods. Various modification have been introduced for Runge-Kutta schemes to attain the purpose. Sonic of them have been tested numerically.The achievement, of the research work has been or is to be published in many articles, a part. of which is listed in the following page.