Efficient simulation of mixed boundary value problems and conformal mappings

Qiansheng Han, Antti Rasila and Tommi Sottinen

Applied Mathematics and Computation, 2025, vol. 488, issue C

Abstract: We present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on-Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an L-shaped insulated polyhedron.

Keywords: Brownian motion; Monte Carlo methods; Walk on spheres; Harmonic measure; Conformal mappings; Conformal modulus; Numerical algorithm; Heat flow (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:488:y:2025:i:c:s0096300324005800

DOI: 10.1016/j.amc.2024.129119

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