Research on infinite precision numerical simulation of partial differential equations

偏微分方程无限精度数值模拟研究

• 批准号: 10354001
• 负责人: IMAI Hitoshi
• 金额: $ 17.22万
• 依托单位: The University of Tokushima
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10354001/
• 关键词: multiple precision; parallel computing; PVM; simulation; spectral method; verification; arbitrary; infinite; 計算機支援; 任意精度; 無限精度; 高精度

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.

在项目实施期间（三年），取得了许多成果。主要结果如下：1.并行计算环境的建立。购买了高性能工作站，并通过高速网络连接。PVM（并行虚拟机）在这些工作站上实现。因此，并行计算环境在Tokusima大学的Imai实验室建立。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。将谱配点法与多精度算法相结合，发展了无限精度数值模拟方法。谱（配点）法用于截断误差的控制。采用多精度算法控制舍入误差。将该方法应用于具有光滑解的偏微分方程组。观察到该方法的特点，即任意减少误差。在一维边值问题中，误差约为10^<-2300>。这与其他数值方法的结果相比是令人难以置信的。库的开发及其发布。开发了无限精度数值模拟库。开发了多精度算法中的高斯消去子程序。利用PVM实现了该库的并行化，并通过在Imai的主页上加载该研究项目的报告发布了该库.相关成果：在可视化无限放大技术的发展、区域分解并行计算技术的发展、无穷高精度数值模拟在反问题、自由边界问题、流体力学中的基础研究与应用、验证研究等方面取得了许多相关成果。

项目成果

M.Tabata: "A precise computation of drag coefficients of a sphere"The International Journal of Computational Fluid Dynamics. Vol.9. 303-311 (1998)

M.Tabata：“球体阻力系数的精确计算”国际计算流体动力学杂志。

Shoichi Fujima: "Mortar element method for flow problems in primitive variables form" International Journal of Computational Fluid Dynamics. vol.9. 209-219 (1998)

Shoichi Fujima：“原始变量形式流动问题的砂浆元法”国际计算流体动力学杂志。

T.Nishida: "Bifurcation problems for equations of fluid dynamics and computer assisted proof"Taiwanese Journal of Mathematics. Vol.4,No.1. 119-127 (2000)

T.Nishida：“流体动力学方程的分岔问题和计算机辅助证明”台湾数学杂志。

H.IMAI (Coauthor : T.TAKEUCHI): "On Numerical Simulation of Partial Differential Equations in Infinite Precision"Advances in Mathematical Sciences and Applications. Vol.9, No.2. 1007-1016 (1999)

H.IMAI（合著者：T.TAKEUCHI）：“无限精度偏微分方程的数值模拟”数学科学与应用的进展。

Pyi Ay (Coauthor : T.Nishida): "Heat convection of compressible fluid, in Recent Developments in Domain Decomposition Methods and Flow Problems"Mathematical Sciences and Applications. Vol.11. 107-115 (1998)

Pyi Ay（合著者：T.Nishida）：“可压缩流体的热对流，域分解方法和流动问题的最新进展”数学科学与应用。