Conformal map and harmonic measure of the Bunimovich stadium

TL;DR

This paper develops a numerical method to compute the conformal map of the Bunimovich stadium and evaluates its harmonic measure, comparing it with other techniques like Schwarz-Christoffel transformations and Monte Carlo simulations.

Contribution

It introduces a Chebyshev-weighted solution of Symm's integral equation for conformal mapping of complex regions with corners.

Findings

The method accurately computes the conformal map of the Bunimovich stadium.

Harmonic measure at the center is effectively estimated and validated.

Comparison shows advantages over traditional Schwarz-Christoffel and Monte Carlo methods.

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.