A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains

Abstract

A simple numerical method is described for computing the following three conformal maps; (a) from a domain interior to a closed Jordan curve onto the interior of the unit disk, (b) from a domain exterior to a closed Jordan curve onto the exterior of the unit disk, and (c) from a doubly-connected domain bounded by two closed Jordan curves onto a circular annulus. The numerical method is based on the charge simulation method or the method of fundamental solutions applied to the Dirichlet problem of Laplace's equation, in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials without integration. Typical examples show that numerical results of high accuracy can be obtained by the simple method if the domains have no concave corners.