A Monte Carlo method for Poisson's equation
Journal of Computational Physics
Extrapolated surface charge method for capacity calculation of polygons and polyhedra
Journal of Computational Physics
Journal of Computational Physics
Algorithm 806: SPRNG: a scalable library for pseudorandom number generation
ACM Transactions on Mathematical Software (TOMS)
ε-Shell error analysis for "Walk On Spheres" algorithms
Mathematics and Computers in Simulation
The random walk on the boundary method for calculating capacitance
Journal of Computational Physics
Capacity of the regular polyhedra
Computers & Mathematics with Applications
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It is well known that there is no analytic expression for the electrical capacitance of the unit cube. However, there are several Monte Carlo methods that have been used to numerically estimate this capacitance to high accuracy. These include a Brownian dynamics algorithm [H.-X. Zhou, A. Szabo, J.F. Douglas, J.B. Hubbard, A Brownian dynamics algorithm for calculating the hydrodynamic friction and the electrostatic capacitance of an arbitrarily shaped object, J. Chem. Phys. 100 (5) (1994) 3821-3826] coupled to the ''walk on spheres'' (WOS) method [M.E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Stat. 27 (1956) 569-589]; the Green's function first-passage (GFFP) algorithm [J.A. Given, J.B. Hubbard, J.F. Douglas, A first-passage algorithm for the hydrodynamic friction and diffusion-limited reaction rate of macromolecules, J. Chem. Phys. 106 (9) (1997) 3721-3771]; an error-controlling Brownian dynamics algorithm [C.-O. Hwang, M. Mascagni, Capacitance of the unit cube, J. Korean Phys. Soc. 42 (2003) L1-L4]; an extrapolation technique coupled to the WOS method [C.-O. Hwang, Extrapolation technique in the ''walk on spheres'' method for the capacitance of the unit cube, J. Korean Phys. Soc. 44 (2) (2004) 469-470]; the ''walk on planes'' (WOP) method [M.L. Mansfield, J.F. Douglas, E.J. Garboczi, Intrinsic viscosity and the electrical polarizability of arbitrarily shaped objects, Phys. Rev. E 64 (6) (2001) 061401:1-061401:16; C.-O. Hwang, M. Mascagni, Electrical capacitance of the unit cube, J. Appl. Phys. 95 (7) (2004) 3798-3802]; and the random ''walk on the boundary'' (WOB) method [M. Mascagni, N.A. Simonov, The random walk on the boundary method for calculating capacitance, J. Comp. Phys. 195 (2004) 465-473]. Monte Carlo methods are convenient and efficient for problems whose solution includes singularities. In the calculation of the unit cube capacitance, there are edge and corner singularities in the charge density distribution. In this paper, we review the above Monte Carlo methods for computing the electrical capacitance of a cube and compare their effectiveness. We also provide a new result. We will focus our attention particularly on two Monte Carlo methods: WOP [M.L. Mansfield, J.F. Douglas, E.J. Garboczi, Intrinsic viscosity and the electrical polarizability of arbitrarily shaped objects, Phys. Rev. E 64 (6) (2001) 061401:1-061401:16; C.-O. Hwang, M. Mascagni, Electrical capacitance of the unit cube, J. Appl. Phys. 95 (7) (2004) 3798-3802; C.-O. Hwang, T. Won, Edge charge singularity of conductors, J. Korean Phys. Soc. 45 (2004) S551-S553] and the random WOB [M. Mascagni, N.A. Simonov, The random walk on the boundary method for calculating capacitance, J. Comp. Phys. 195 (2004) 465-473] methods.