Plasma and Fusion Research

Volume 17, 1403029 (2022)

Regular Articles

Monte Carlo Solver for Partly Calculating a Solution to the Poisson Equation in Three-Dimensional Curvilinear Coordinates
Ryutaro KANNO1,2), Gakushi KAWAMURA1,2), Masanori NUNAMI1,3), Seikichi MATSUOKA1,2) and Shinsuke SATAKE1,2)
National Institute for Fusion Science, National Institutes of Natural Sciences, Toki 509-5292, Japan
Department of Fusion Science, The Graduate University for Advanced Studies, SOKENDAI, Toki 509-5292, Japan
Nagoya University, Nagoya 464-8601, Japan
(Received 21 November 2021 / Accepted 19 February 2022 / Published 22 April 2022)


We develop a new simulation code for solving the Poisson equation, based on Monte Carlo methods. When static resonant magnetic perturbations (RMPs) are used in tokamak plasma to mitigate or suppress edge-localized modes, the RMPs generate an electric field in the ergodized edge region. The electrostatic potential should be calculated only in the edge region to reduce the computational cost of solving the Poisson equation in the complicated three-dimensional magnetic structure, which is assumed to be fixed in time. In this study, we propose a basic idea for evaluating an electrostatic potential given by the Poisson equation in only a part of the domain in curvilinear coordinates. This Poisson solver allows for the boundary condition to be set not only inside the selected region in which the potential is evaluated, but also outside the selected region. Several benchmarks for the developed code are also presented.


Poisson solver, Monte Carlo method, curvilinear coordinate

DOI: 10.1585/pfr.17.1403029


  • [1] T. Evans et al., Phys. Rev. Lett. 92, 235003 (2004).
  • [2] B.S. Victor et al., Plasma Phys. Control. Fusion 62, 095021 (2020).
  • [3] S. Mordijck et al., Nucl. Fusion 54, 082003 (2014).
  • [4] O. Schmitz et al., Plasma Phys. Control. Fusion 50, 124029 (2008).
  • [5] W.H. Press et al., Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University Press, Cambridge, 2007).
  • [6] F. Zhang et al., Eng. Comput. 32, 285 (2006).
  • [7] T. Moritaka et al., Plasma 2, 179 (2019).
  • [8] J.M. DeLaurentis and L.A. Romero, J. Comput. Phys. 90, 123 (1990).
  • [9] K. Itô, The Brownian motion and tensor fields on Riemannian manifold, in Proceedings of the International Congress of Mathematicians (International Congress of Mathematicians, 15 - 22 August 1962, Stockholm), p.536-539.
  • [10] R. Kanno et al., Plasma Fusion Res. 6, 2403066 (2011).
  • [11] B. Øksendal, Stochastic Differential Equations (Springer-Verlag, Berlin Heidelberg, 2003).
  • [12] A. Friedman, Stochastic Differential Equations and Applications (Dover, New York, 2004).
  • [13] M. Endo, (in Japanese).
  • [14] J.R. Klauder and W.P. Petersen, SIAM J. Numer. Anal. 22, 1153 (1985).
  • [15] T. Misawa and H. Itakura, Phys. Rev. E 51, 254 (1995).
  • [16] K. Harafuji et al., J. Comput. Phys. 81, 169 (1989).
  • [17] R.E. Hiromoto and R.G. Brickner, Empirical Results of a Hybrid Monte Carlo Method for the Solution of Poisson's Equation, In: Brebbia C.A., Peters A., Howard D. (eds) Applications of Supercomputers in Engineering II. (Springer, Dordrecht, 1991) p.233-239.