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

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，电荷模拟方法是拉普拉斯方程狄利克雷问题的一种简单、准确的求解方法。我们提出了Nehari （Mc-Graw Hill, 1952）正则域(a)平行缝域，(b)圆缝域，(c)径向缝域，（d)具有同心圆缝的圆和(e)具有同心圆缝的圆环）上多重连通域的数值共形映射的一种简单方法。该方法采用复平面上的电荷模拟方法，即复对数函数的线性组合，在边界曲线和边界数据是解析性的情况下，给出了精度较高的近似映射函数。我们成功地将数值保形映射应用到势流分析中，并给出了求障碍物周围滞止点和计算障碍物作用力的简单方法。我们提出了一些将电荷模拟方法应用于有角或狭缝的畴的技术。我们还提出了一种适用于周期域的新型电荷模拟方法，该方法使用问题的周期基本解来代替对数函数。证明了在电荷模拟方法的不变格式中出现的线性系统的唯一可解性。这些结果使使用基本解的经典方法在计算机时代成为一种现代方法。

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.