Simulation Notes Europe, Volume 30(2), June 2020

CTDS – Mauling Heat Equation on Unit Disk by Conformal Parametrization

Simulation Notes Europe SNE 30(2), 2020, 43-50
DOI: 10.11128/


Let us consider a heat conduction problem on the unit square, solve it analytically and compare this series solution with both the results obtained by MATLAB/PDE-Toolbox (using the FE-method) and a selfimplementation achieved with Mathematica (using the method of lines). Extending Finite-Difference formulae to higher precision gives rise to the idea of utilizing the CTDS-method, best suitable on regular and equidistant grids, also on other domains. By introducing apt coordinates one is therefore able to do a parametrization, e.g. of the unit disk, by the square. Conformal transformation from square to disk provides this parametrization, the original implementation can easily be extended and we find by again comparing a series solution to our obtained simulation results, that order of convergence is being preserved. Moreover, our conformal transformation provides the fundamental tensor and no further structural errors are being introduced as the involved elliptic functions can be evaluated to arbitrary precision.