Conformal map and harmonic measure of the Bunimovich stadium

Abstract

We consider the conformal mapping of the Bunimovich stadium, a region enclosed by a Jordan curve with four smooth corners, primarily in the context of a particle undergoing Brownian motion within its closed geometry with Dirichlet boundary conditions. A Chebyshev weighting of the solutions of Symm's integral equation is employed to give a numerical conformal map of the region onto the canonical domain of the unit disk in the complex plane. As a measure of the accuracy of the transformation, the domes' harmonic measure evaluated at the centre of the stadium is thereby extracted and is compared with results obtained from Schwarz-Christoffel transformations and Monte Carlo simulations; the pros and cons of the method are reiterated.