High Precision Conformal Map of the Unit Disk to Star-Shaped Domains

Having the conformal map from unit square to unit disk at hand, we ask ourselves for a way to numerically map the disk to more general domains. Once such domains are parametrized by the square, knowledge of metric quantities flows in by the derivatives of the map and simulation of PDEs on such domains can easily be achieved. One way to numerically construct the map for star-shaped regions yields over solving Theodorsen’s integral equation which establishes the boundary correspondence of angles. Focusing on highly accurate solutions, we presentMathematica test implementations and results for maps from unit disk to inverted ellipse (a), unit square (b) and onto a more general domain (c). Finally, the metric impact of the conformal map on the PDE itself is being investigated to enlighten the process of correcting spacial Finite-Difference approximations in general.