Research on infinite precision numerical simulation of partial differential equations

在项目实施期间(三年)，取得了许多成果。主要研究成果如下：1.本文的主要研究成果。建立并行计算环境。购买高性能工作站，并通过高速网络连接。在这些工作站上实现了PVM(并行虚拟机)。因此，并行计算环境是在今井在德意志大学的实验室建立的。无限精度数值模拟的发展数值模拟中的误差来源于离散化中的截断误差和舍入误差。将谱(配)法与多精度算法相结合，发展了无限精度数值模拟。采用谱(配置法)控制截断误差。采用多精度算法对舍入误差进行控制。将该方法应用于具有光滑解的偏微分方程组。观察到了该方法的特点，即任意减少误差。在一维边值问题中，误差约为10^-2300；与其他数值方法的结果相比，这是令人难以置信的。开发了无限精度数值模拟程序库及其发布版本。开发了多精度运算中的高斯消去子程序。利用PVM进行并行化，通过在Imai主页上下载研究项目报告来发布文库。在可视化中无限放大化的发展、区域分解并行计算的发展、无限精度数值模拟在反问题、自由边界问题和流体力学中的基础研究和应用、验证研究等方面取得了许多相关成果。

In the term of the project (three years) many results were obtained. Important results are shown as follows.1. Building of the parallel computing environment.High-performance workstations were purchased and connected by the high-speed network. PVM (Parallel Virtual Machine) was implemented on these workstations. Thus, the parallel computing environment was built at Imai's laboratory at Tokusima University.2. Development of Infinite Precision Numerical Simulation.Errors in numerical simulations originate from truncation errors in discretization and rounding errors. Infinite Precision Numerical Simulation was developed by combining the spectral (collocation) method and multiple precision arithmetic. The spectral (collocation) method is used for the control of truncation errors. Multiple precision arithmetic is used for the control of rounding errors. The method was applied to PDE systems with smooth solutions. The feature of the method, i.e. arbitrary reduction of errors, was observed. In a one-dimensional boundary value problem errors were approximately 10^<-2300>. This is incredible compared with results by other numerical methods.3. Development of the library and its release.The library for Infinite Precision Numerical Simulation was developed. The subroutine of Gauss elimination in multiple precision arithmetic was developed. Its parallelization was performed by using PVM.The library was released by up-loading the report of the research project on Imai's home page.4. Related results.Many related results were obtained as for development of infinite magnifying in visualization, development of parallel computing by using domain decomposition, basic research and application of Infinite Precision Numerical Simulation to inverse problems, free boundary problems and fluid mechanics, research on verification.