Numerical conformal mapping of circular arc polygons☆

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Abstract

Howell, L.H., Numerical conformal mapping of circular arc polygons, Journal of Computational and Applied Mathematics 46 (1993) 7–28.

Solutions to the Schwarzian differential equation are conformal maps from the upper half-plane to circular arc polygons, plane regions bounded by straight line segments and arbitrary arcs of circles. We develop methods for numerically integrating this equation, both directly and through the use of a related linear differential equation. Particular attention is given to the behavior near corner singularities. We also derive alternate versions of the transformation which map from the unit disk and from an infinite strip. While the former may be of primarily theoretical interest, the latter can be used to map highly elongated regions such as channels for internal flow problems. Such regions are difficult or impossible to map from the disk or the half-plane due to the so-called crowding phenomenon.

Keywords

Schwarz-Christoffel transformation
Schwarzian
ordinary differential equation
elongated regions
crowding
conformal mapping

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Portions of this work were performed at the Massachusetts Institute of Technology under the support of U.S. Air Force Grant AFOSR-87-0102. Other parts of the work were performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Partial support under contract No. W-7405-Eng-48 was provided by the Applied Mathematical Sciences Program of the Office of Energy Research.

Copyright © 1993 Published by Elsevier B.V.