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A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. (English) Zbl 0818.30004
It is well known that the conformal mapping of a simply [or doubly] connected domain $D$ onto the unit disk [or a circular ring] can be reduced to the solution of a Dirichlet problem. The author solves the latter by the ‘charge simulation method’ which has been studied extensively by several Japanese authors: Murashima, Katsurada, Okamoto and the present author. In this method the solution of the Dirichlet problem is approximated by $G\left(z\right)={\sum }_{i=1}^{N}{Q}_{i}log|z-{\zeta }_{i}|$ with unknown coefficients ${Q}_{i}$, where the charge points ${\zeta }_{i}$ lie outside $\overline{D}$. To determine the ${Q}_{i}$ it is required that $G\left({z}_{j}\right)=b\left({z}_{j}\right)$ for $N$ collocation points ${z}_{j}\in \partial D$. The error is defined by ${E}_{G}=max|G\left({z}_{j+\frac{1}{2}}\right)-b\left({z}_{j+\frac{1}{2}}\right)|$, where ${z}_{j+\frac{1}{2}}\in \partial D$ are intermediate points. The determination of $G$ thus leads to the solution of a $N×N$ linear system for the ${Q}_{i}$. – The paper fully covers earlier work and gives a unified treatment of three types of mapping problems. The choice of the charge points is discussed, and the condition of the $N×N$ matrix is studied. Results of 11 numerical examples are given and compared with previous experiments. Good results are reported in all cases provided that no concave corners on $\partial D$ occur.
##### MSC:
 30C30 Numerical methods in conformal mapping theory
##### Keywords:
charge simulation method