Applications of Mathematics

, Volume 64, Issue 1, pp 33–43 | Cite as

Unique solvability and stability analysis of a generalized particle method for a Poisson equation in discrete Sobolev norms

  • Yusuke ImotoEmail author


Unique solvability and stability analysis is conducted for a generalized particle method for a Poisson equation with a source term given in divergence form. The general- ized particle method is a numerical method for partial differential equations categorized into meshfree particle methods and generally indicates conventional particle methods such as smoothed particle hydrodynamics and moving particle semi-implicit methods. Unique solv- ability is derived for the generalized particle method for the Poisson equation by introducing a connectivity condition for particle distributions. Moreover, stability is obtained for the discretized Poisson equation by introducing discrete Sobolev norms and a semi-regularity condition of a family of discrete parameters.


generalized particle method Poisson equation unique solvability stability discrete Sobolev norm 

MSC 2010



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. Ben Moussa: On the convergence of SPH method for scalar conservation laws with boundary conditions. Methods Appl. Anal. 13 (2006), 29–61.MathSciNetzbMATHGoogle Scholar
  2. [2]
    B. Ben Moussa, J. P. Vila: Convergence of SPH method for scalar nonlinear conservation laws. SIAM J. Numer. Anal. 37 (2000), 863–887.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S. J. Cummins, M. Rudman: An SPH projection method. J. Comput. Phys. 152 (1999), 584–607.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. A. Gingold, J. J. Monaghan: Smoothed particle hydrodynamics: theory and applica-tion to non-spherical stars. Mon. Not. R. Astron. Soc. 181 (1977), 375–389.CrossRefzbMATHGoogle Scholar
  5. [5]
    Y. Imoto: Error estimates of generalized particle methods for the Poisson and heat equa-tions. Ph. D. Thesis, Kyushu University Institutional Repository, Fukuoka, 2016.Google Scholar
  6. [6]
    Y. Imoto, D. Tagami: A truncation error estimate of the interpolant of a particle method based on the Voronoi decomposition. JSIAM Lett. 8 (2016), 29–32.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Y. Imoto, D. Tagami: Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition. JSIAM Lett. 9 (2017), 69–72.MathSciNetCrossRefGoogle Scholar
  8. [8]
    K. Ishijima, M. Kimura: Truncation error analysis of finite difference formulae in mesh-free particle methods. Trans. Japan Soc. Ind. Appl. Math. 20 (2010), 165–182. (In Japanese. )Google Scholar
  9. [9]
    S. Koshizuka, Y. Oka: Moving-particle semi-implicit method for fragmentation of incom-pressible fluid. Nuclear Sci. Eng. 123 (1996), 421–434.CrossRefGoogle Scholar
  10. [10]
    L. B. Lucy: A numerical approach to the testing of the fission hypothesis. Astronom. J. 82 (1977), 1013–1024.CrossRefGoogle Scholar
  11. [11]
    P.-A. Raviart: An analysis of particle methods. Numerical Methods in Fluid Dynamics (F. Brezzi et al., eds. ). Lecture Notes in Math. 1127, Springer, Berlin, 1985, pp. 243–324.zbMATHGoogle Scholar
  12. [12]
    S. Shao, E. Y. M. Lo: Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. Water Resources 26 (2003), 787–800.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2019

Authors and Affiliations

  1. 1.Tohoku Forum for CreativityTohoku UniversitySendaiJapan

Personalised recommendations