Accurate Implementation of the Schwarz-Christoffel Tranformation

Evan Warner

What is it?

A conformal mapping (preserves angles and infinitesimal shapes) that maps polygons onto a simpler domain in the complex plane

Amazing Riemann Mapping theorem: A conformal (analytic and bijective) map always

exists for a simply connected domain to the unit circle, but it doesn't say how to find it

Schwarz-Christoffel formula is a way to take a certain subset of simply connected domains (polygons) to find the necessary mapping

Why does anyone care?

Physical problems: Laplace's equation, Poisson's equation, the heat equation, fluid flow and others on polygonal domains

To solve such a problem: State problem in original domain Find Schwarz-Christoffel mapping to simpler

domain Transform differential equation under mapping Solve Map back to original domain using inverse

transformation (relatively easy to find)

Who has already done this?

Numerical methods, mostly in FORTRAN, have existed for a few decades

Various programs use various starting domains, optimizations for various polygon shapes

Long, skinny polygons notoriously difficult, large condition numbers in parameter problem

Continuous Schwarz-Christoffel problem, involving integral equation instead of discrete points, has not been successfully implemented

How to find a transformation...

State the domain, find the angles of the polygon, and come up with the function given by the formula:

http://math.fullerton.edu/mathews/c2003/SchwarzChristoffelMod.html

B and A are constants determined by the solution to the parameter problem, the x's are the points of the original domains, the alphas are the angles

How to find a transformation...

Need a really fast, accurate method of computing that integral (need numerical methods) many many times.

Gauss-Jacobi quadrature provides the answer: quadrature routine optimized for the necessary weighting function.

Necessary to derive formulae for transferring the idea to the complex domain.

How to find a transformation...

The parameter problem must be solved – either of two forms, constrained linear equations or unconstrained nonlinear equations (due to Trefethen)

Solve for prevertices - points along simple domain that map to verticies

Once prevertices are found, transformation is found

Examples

Upper half-plane to semi-infinite strip; lines are Re(z)=constant and Im(z)=constant

Examples

Mapping from upper half-plane to unit square; lines are constant for the opposite image

What have I done so far?

Implementation of complex numbers in java ComplexFunction class Implementation of Gauss-Jacobi quadrature Basic graphical user interface with capability to

calculate Gauss-Jacobi integrals Testing done mostly in MATLAB (quad routine)

What's next?

Research into solving the nonlinear system parameter problem – compare numerical methods

Independent testing program for a variety of domains, keeping track of mathematically computed maximum error bounds

User-friendly GUI for aids in solving physical problems and equations