Accurate Implementation of the Schwarz-Christoffel Tranformation

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Accurate Implementation of the Schwarz-Christoffel
Tranformation

• Evan Warner

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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

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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)

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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

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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/SchwarzChr
istoffelMod.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
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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.

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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

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

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

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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)

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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