An energy stable incompressible SPH method with consistent solid boundary treatment

Xingyu Zhu, Xiuping Wang, Jisheng Kou, Shuyu Sun*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We propose two novel solid wall boundary treatments on the energy-stable Smoothed Particle Hydrodynamics (SPH) method for the Navier–Stokes system which models single-phase incompressible flows. In this work, the solid wall is discretized with dummy particles, and an external force is introduced to restrict the motion of solid particles, whose physical significance and numerical necessity are analyzed. We construct consistent fluid–solid coupling schemes based on the approaches so that the boundary condition is involved in the same equations system with particular constraints. Furthermore, the schemes are unconditionally energy-stable, which allows large time step sizes and leads to efficient computation. A variety of examples are demonstrated to verify the corresponding numerical analysis result, which shows that the approaches can easily handle no-slip and moving boundaries. Finally, we highlight that the energy-stable ISPH with consistent boundary treatments greatly benefits solving long-term problems with steady states.

Original languageEnglish (US)
Article number115367
JournalJournal of Computational and Applied Mathematics
StatePublished - Jan 15 2024

Bibliographical note

Funding Information:
This work is partially supported by King Abdullah University of Science and Technology (KAUST), Saudi Arabia through the grants BAS/1/1351-01 , URF/1/4074-01 , and URF/1/3769-01 .

Publisher Copyright:
© 2023


  • Consistent boundary treatments
  • Energy stability
  • Incompressible smoothed particle hydrodynamics (SPH)
  • Numerical stability
  • Solid wall boundary

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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