Computation of Conformal Maps by Modiied Schwarz-christooel Transformations Certiied by Accepted by Computation of Conformal Maps by Modiied Schwarz-christooel Transformations

Numerical conformal mapping techniques can be roughly divided into two categories. On one hand there are many methods which perform well on regions with smooth boundaries , but which either fail or require elaborate modiications when mapping regions with corners. On the other hand there is the Schwarz-Christooel transformation, which is speciically designed for mapping polygons with any number of corners, but which does not, at least in its original form, extend to mapping domains with curved sides. In this thesis we explore four issues regarding the use of the Schwarz-Christooel transformation and related methods as robust conformal mapping techniques. The rst of these, the crowding phenomenon, makes maps onto even mildly elongated regions so ill-conditioned they may be impossible to calculate. We overcome this diiculty by implementing a transformation, similar to the Schwarz-Christooel formula, which uses an innnite strip as the standard domain. The mapping function itself is not original; it has previously been used by others to compute maps onto open channels for internal ow computations. The interpretation in terms of crowding is original, however, and we also present a new modiication for mapping elongated rectangles onto closed polygons. By using robust and eecient software for numerical integration and solution of the parameter problem, we successfully map diicult regions with aspect ratios of up to thousands to one. Second, we survey a number of techniques for numerically evaluating Schwarz-Christooel integrals, including the compound Gauss-Jacobi algorithm, singularity removal methods, and several adaptive quadrature subroutines. The Gauss-Jacobi method is shown to be faster than its nearest competitor by roughly a factor of two; we recommend it for all applications unless strict error bounds are required. Experimental data are presented to support our conclusions. Third, all Schwarz-Christooel methods require the solution of a system of nonlinear algebraic equations. We examine two approaches to this problem that have appeared 2 in the literature, Davis's iterative scheme and Trefethen's formulation as a nonlinear system. We show that Davis's method is diicult to generalize and is not always even locally convergent, and prove that although Trefethen's method never diverges locally, it too may diverge from distant starting points. We then discuss several approaches to the problem of designing a globally convergent algorithm. Finally, a generalization of the Schwarz-Christooel formula, dating back to Schwarz in 1869, describes conformal maps onto regions bounded by arbitrary arcs of circles as solutions to a nonlinear ordinary diierential equation. …