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Accurate Implementation of the Schwarz-Christoffel Tranformation

Evan Warner. Accurate Implementation of the Schwarz-Christoffel Tranformation. 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:

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Accurate Implementation of the Schwarz-Christoffel Tranformation

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  1. Evan Warner Accurate Implementation of the Schwarz-Christoffel Tranformation

  2. 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

  3. 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)

  4. 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

  5. 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

  6. 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.

  7. 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

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

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

  10. 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)

  11. 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

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