Computational and Applied Mathematics

, Volume 35, Issue 2, pp 447–473 | Cite as

Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations

  • Paul MycekEmail author
  • Grégory Pinon
  • Grégory Germain
  • Elie Rivoalen


The modelling of diffusive terms in particle methods is a delicate matter and several models were proposed in the literature to take such terms into account. The diffusion velocity method (DVM), originally designed for the diffusion of passive scalars, turns diffusive terms into convective ones by expressing them as a divergence involving a so-called diffusion velocity. In this paper, DVM is extended to the diffusion of vectorial quantities in the three-dimensional Navier–Stokes equations, in their incompressible, velocity–vorticity formulation. The integration of a large eddy simulation (LES) turbulence model is investigated and a DVM general formulation is proposed. Either with or without LES, a novel expression of the diffusion velocity is derived, which makes it easier to approximate and which highlights the analogy with the original formulation for scalar transport. From this statement, DVM is then analysed in one dimension, both analytically and numerically on test cases to point out its good behaviour.


Diffusion velocity method Particle method Fourier analysis  Transport-dispersion equations Navier–Stokes equations 

Mathematics Subject Classification

76M23 76M28 76R50 76D05 76F65 65M75 



The authors would like to thank the Haute-Normandie Regional Council and the Institut Français de Recherche pour l’Exploitation de la Mer (IFREMER) for their financial support of co-financed PhD theses, as well as the Réseau d’Hydrodynamique Normand (RHYNO). The authors would also like to thank the Centre des Ressources Informatiques de HAute-Normandie (CRIHAN) for their available numerical computation resources.


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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014

Authors and Affiliations

  • Paul Mycek
    • 1
    Email author
  • Grégory Pinon
    • 2
  • Grégory Germain
    • 3
  • Elie Rivoalen
    • 2
    • 4
  1. 1.Department of Mechanical Engineering and Materials ScienceDuke UniversityDurhamUSA
  2. 2.Laboratoire Ondes et Milieux Complexes, UMR 6294CNRS-Université du HavreLe Havre CedexFrance
  3. 3.IFREMER, Hydrodynamic and Metocean ServiceBoulogne-Sur-MerFrance
  4. 4.Laboratoire d’Optimisation et Fiabilité en Mécanique des Structures, EA 3828INSA de RouenSaint-Etienne-du-RouvrayFrance

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