Reproducing kernel particle methods

Wing Kam Liu*, Sukky Jun, Yi Fei Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2521 Scopus citations


A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time‐frequency and space‐wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element‐free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C, the solution and its derivatives are also C in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two‐dimensional examples of advection‐diffusion equations and compressible Euler equations are also presented together with 2D multiple‐scale decompositions.

Original languageEnglish (US)
Pages (from-to)1081-1106
Number of pages26
JournalInternational Journal for Numerical Methods in Fluids
Issue number8-9
StatePublished - 1995


  • correction function
  • mesh‐ (or grid‐) free particle methods
  • multiple scale decomposition
  • multi‐resolution analysis
  • reproducing kernel function
  • wavelet

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Computer Science Applications
  • Applied Mathematics


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