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General Research on the Charge Simulation Method and the Numerical Conformal Mapping

电荷模拟方法与数值共角映射的综合研究

基本信息

批准号：
12440029
负责人：
AMANO Kaname
金额：
$ 5.25万
依托单位：
Ehime University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440029/
关键词：
charge simulation method Laplace's equation fundamental solution numerical conformal mapping multiply connected domain potential problem complex analysis numerical analysis ボテンシャル問題

项目摘要

The numerical conformal mapping has been an important subject in computational mathematics. On the other hand, the charge simulation method is a simple accurate solver for the Dirichlet problem of the Laplace equation.1.We have proposed a simple method of numerical conformal mappings of multiply-connected domains onto the canonical domains of Nehari (Mc-Graw Hill, 1952), i.e., (a) the parallel slit domain, (b) the circular slit domain, (c) the radial slit domain, (d) the circle with concentric circular slits and (e) the circular ring with concentric circular slits. The method uses the charge simulation method on the complex plane, i.e., a linear combination of complex logarithmic functions, and gives approximate mapping functions with high accuracy if boundary curves and boundary data are analytic.2.We have successfully applied the numerical conformal mapping to potential flow analysis, and presented simple methods to find the stagnation points around obstacles and to compute the forces on obstacles.3.We have presented some techniques to apply the charge simulation method to domains with corners or slits. We also presented new types of the charge simulation method applicable to periodic domains, which use periodic fundamental solutions of the problem instead of logarithmic functions.4.We proved the unique solvability of the linear systems appearing in the invariant scheme of the charge simulation method.These results revive the classical method using fundamental solutions as a modern method in the computer age.

数值保角映射一直是计算数学中的一个重要课题。另一方面，电荷模拟法是求解拉普拉斯方程Dirichlet问题的一种简单精确的方法。1.我们提出了一种简单的多重连通区域到Nehari正则区域的数值共形映射方法(Mc-Graw Hill，1952)，即(A)平行狭缝区域，(B)圆形狭缝区域，(C)径向狭缝区域，(D)带同心圆缝的圆和(E)带同心圆缝的圆环。该方法使用复平面上的电荷模拟方法，即复对数函数的线性组合，在分析边界曲线和边界数据的情况下，给出了高精度的近似映射函数。2.成功地将数值保角映射应用到势流分析中，并给出了寻找障碍物周围的停滞点和计算障碍物上的力的简单方法。我们还提出了适用于周期域的新型电荷模拟方法，它使用问题的周期基本解来代替对数函数。4.证明了电荷模拟方法不变格式中出现的线性系统的唯一可解性，这些结果使利用基本解作为现代方法的经典方法在计算机时代复兴。