Unique solvability and stability analysis for incompressible smoothed particle hydrodynamics method

  • Yusuke ImotoEmail author


The incompressible smoothed particle hydrodynamics (ISPH) method is a numerical method widely used for accurately and efficiently solving flow problems with free surface effects. However, to date there has been little mathematical investigation of properties such as stability or convergence for this method. In this paper, unique solvability and stability are mathematically analyzed for implicit and semi-implicit schemes in the ISPH method. Three key conditions for unique solvability and stability are introduced: a connectivity condition with respect to particle distribution and smoothing length, a regularity condition for particle distribution, and a time step condition. The unique solvability of both the implicit and semi-implicit schemes in two- and three-dimensional spaces is established with the connectivity condition. The stability of the implicit scheme in two-dimensional space is established with the connectivity and regularity conditions. Moreover, with the addition of the time step condition, the stability of the semi-implicit scheme in two-dimensional space is established. As an application of these results, modified schemes are developed by redefining discrete parameters to automatically satisfy parts of these conditions.


Incompressible smoothed particle hydrodynamics method Incompressible Navier–Stokes equations Unique solvability Stability 


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Conflict of interest

The author declares no conflicts of interest.


  1. 1.
    Asai M, Aly AM, Sonoda Y, Sakai Y (2012) A stabilized incompressible SPH method by relaxing the density invariance condition. J Appl Math, 139583Google Scholar
  2. 2.
    Ben Moussa B (2006) On the convergence of SPH method for scalar conservation laws with boundary conditions. Methods Appl Anal 13(1):29–62MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ben Moussa B, Vila J (2000) Convergence of SPH method for scalar nonlinear conservation laws. SIAM J Numer Anal 37(3):863–887MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chorin AJ (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22(104):745–762MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cummins SJ, Rudman M (1999) An SPH projection method. J Comput Phys 152(2):584–607MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics-theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389CrossRefzbMATHGoogle Scholar
  7. 7.
    Gresho PM (1990) On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: theory. Int J Numer Methods 11(5):587–620MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Imoto Y (2016) Error estimates of generalized particle methods for the Poisson and heat equations. Ph.D. thesis, Kyushu UniversityGoogle Scholar
  9. 9.
    Lind S, Stansby P (2016) High-order Eulerian incompressible smoothed particle hydrodynamics with transition to Lagrangian free-surface motion. J Comput Phys 326:290–311MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lind S, Xu R, Stansby P, Rogers BD (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. J Comput Phys 231(4):1499–1523MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013–1024CrossRefGoogle Scholar
  12. 12.
    Morris JP, Fox PJ, Zhu Y (1997) Modeling low Reynolds number incompressible flows using SPH. J Comput Phys 136(1):214–226CrossRefzbMATHGoogle Scholar
  13. 13.
    Raviart PA (1985) An analysis of particle methods. In: Numerical methods in fluid dynamics (Como, 1983). Lecture Notes in Mathematics, vol 1127. Springer, BerlinGoogle Scholar
  14. 14.
    Shao S, Lo EY (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26(7):787–800CrossRefGoogle Scholar
  15. 15.
    Xu R, Stansby P, Laurence D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. J Comput Phys 228(18):6703–6725MathSciNetCrossRefzbMATHGoogle Scholar

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© OWZ 2018

Authors and Affiliations

  1. 1.Tohoku Forum for CreativityTohoku UniversitySendaiJapan

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