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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)= i=1 N Q i log|z-ζ i | with unknown coefficients Q i , where the charge points ζ i lie outside D ¯. To determine the Q i it is required that G(z j )=b(z j ) for N collocation points z j D. The error is defined by E G =max|G(z j+1 2 )-b(z j+1 2 )|, where z j+1 2 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 D occur.
30C30Numerical methods in conformal mapping theory