Mathematics and Computers in Simulation 63 (2003) 93–104 ǫ-Shell error analysis for “Walk On Spheres” algorithms Michael Mascagni a , Chi-Ok Hwang b,∗,1 a Department of Computer Science, Florida State University, 203 Love Building Tallahassee, FL 32306-4530, USA b Innovative Technology Center for Radiation Safety, Hanyang University, HIT Building, 17 Haengdang-Dong, Sungdong-Gu, Seoul 133-791, South Korea Received 16 November 2002; received in revised form 16 November 2002; accepted 25 February 2003 Abstract The “Walk On Spheres” (WOS) algorithm and its relatives have long been used to solve a wide variety of boundary value problems [Ann. Math. Stat. 27 (1956) 569; J. Heat Transfer 89 (1967) 121; J. Chem. Phys. 100 (1994) 3821; J. Appl. Phys. 71 (1992) 2727]. All WOS algorithms that require the construction of random walks that terminate, employ an ǫ-shell to ensure their termination in a finite number of steps. To remove the error related to this ǫ-shell, Green’s function first-passage (GFFP) algorithms have been proposed [J. Chem. Phys. 106 (1997) 3721] and used in several applications [Phys. Fluids A 12 (2000) 1699; Monte Carlo Meth. Appl. 7 (2001) 213; The simulation–tabulation method for classical diffusion Monte Carlo, J. Comput. Phys. submitted]. One way to think of the GFFP algorithm is as an ǫ = 0 extension of WOS. Thus, an important open question in the use of GFFP is to understand the tradeoff made in the efficiency of GFFP versus the ǫ-dependent error in WOS. In this paper, we present empirical evidence and analytic analysis of the ǫ-shell error in some simple boundary value problems for the Laplace and Poisson equations and show that the error associated with the ǫ-shell is O(ǫ), for small ǫ. This fact supports the conclusion that GFFP is preferable to WOS in cases where both are applicable. © 2003 IMACS. Published by Elsevier Science B.V. All rights reserved. PACS: 87.15.Vv; 84.37.+q; 82.20.Pm Keywords: Walk On Spheres (WOS); Error; Laplace; Poisson 1. Introduction In general, Monte Carlo random walk algorithms inside a domain [5] can be classified into two broad categories: random walks on grids or other discrete objects [6] and continuous, grid-free, ran- dom walks/Brownian motions. Among these Monte Carlo algorithms, the grid-free “Walk On Spheres” ∗ Corresponding author. Tel.: +82-2-22918154; fax: +82-2-22968154. E-mail address: chwang@itrs.hanyang.ac.kr (C.-O. Hwang). 1 Present address: Computational Electronics Center, Inha University, 253 Yonghyeon-dong, Nam-gu, Incheon 402-751, South Korea. 0378-4754/03/$ – see front matter © 2003 IMACS. Published by Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-4754(03)00038-7