Numerical Computation of the Schwarz-Christoffel Transformation.

A program is described which computes Schwartz-Christoffel transformations that map the unit disk conformally onto the interior of a bounded or unbounded polygon in the complex plane. The inverse map is also computed. The computational problem is approached by setting up a nonlinear system of equations whose unknowns are essentially the accessory parameters z sub k. This system is then solved with a packaged subroutine. New features of this work include the evaluation of integrals within the disk rather than along the boundary, making possible the treatment of unbounded polygons the use of a compound form of Gauss-Jacobi quadrature to evaluate the Schwarz-Christoffel integral, making possible high accuracy at reasonable cost and the elimination of constraints in the nonlinear system by a simple change of variables. Schwarz-Christoffel transformations may be applied to solve the Laplace and Poisson equations and related problems in two-dimensional domains with irregular or unbounded but not curved or multiply connected geometries. Computational examples are presented. The time required to solve the mapping problem is roughly proportional to N-cubed, where N is the number of vertices of the polygon. A typical set of computations to 8-place accuracy with N or 10 takes 1 to 10 seconds on an IBM 370168. Author