Smoothed particle hydrodynamics (SPH) is a popular particle- and kernel-based method for numerically computing solutions to fluid-flow problems. In this paper, we study the convergence of SPH for the Euler equations of a specific barotropic fluid. In contrast to previous works, we distinguish carefully between the smoothing and the discretization parameters and give explicit relations between both of them to guarantee convergence. This convergence heavily depends on certain properties of the underlying kernel. Hence, a large part of this paper is devoted to constructing compactly supported, radial kernels which possess the required properties.


  1. Euler equations
  2. smoothed particle hydrodynamics
  3. convergence analysis
  4. kernel-based approximation

MSC codes

  1. 65M15
  2. 35Q31
  3. 65M75
  4. 76M28
  5. 41A30

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4752 - 4784
ISSN (online): 1095-7154


Submitted: 20 November 2017
Accepted: 20 June 2018
Published online: 6 September 2018


  1. Euler equations
  2. smoothed particle hydrodynamics
  3. convergence analysis
  4. kernel-based approximation

MSC codes

  1. 65M15
  2. 35Q31
  3. 65M75
  4. 76M28
  5. 41A30



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : WE 2333/9-1

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