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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 $G(z)= \sum\sp N\sb{i= 1} Q\sb i\log \vert z- \zeta\sb i\vert$ with unknown coefficients $Q\sb i$, where the charge points $\zeta\sb i$ lie outside $\overline D$. To determine the $Q\sb i$ it is required that $G(z\sb j)= b(z\sb j)$ for $N$ collocation points $z\sb j\in \partial D$. The error is defined by $E\sb G= \max\vert G(z\sb{j+{1\over 2}})- b(z\sb{j+{1\over 2}})\vert$, where $z\sb{j+{1\over 2}}\in \partial D$ are intermediate points. The determination of $G$ thus leads to the solution of a $N\times N$ linear system for the $Q\sb i$. -- 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 $N\times N$ 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
Full Text:
References:
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