Walk On Spheres Algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence



We show that a constant-potential time-independent Schrödinger equation with Dirichlet boundary data can be reformulated as a Laplace equation with Dirichlet boundary data. With this reformulation, which we call the Duffin correspondence, we provide a classical Walk On Spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the boundary value problem. We compare the obtained Duffin WOS algorithm with existing modified WOS algorithms.


Brownian motion Helmholtz equation Linearized Poisson–Boltzmann equation Monte Carlo simulation Numerical algorithm Walk On Spheres algorithm Yukawa equation 

Mathematics Subject Classification (2010)

65C05 68U20 35Q40 


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsHunan First Normal UniversityChangshaPeople’s Republic of China
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityEspooFinland
  3. 3.Faculty of Technology, Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland

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