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 Mycek
  • 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.


  1. Beale JT, Majda A (1985) High order accurate vortex methods with explicit velocity kernels. J Comput Phys 58(2): 188–208, doi: 10.1016/0021-9991(85)90176-7,
  2. Beaudoin A, Huberson S, Rivoalen E (2003) Simulation of anisotropic diffusion by means of a diffusion velocity method. J Comput Phys 186(1): 122–135, doi: 10.1016/S0021-9991(03)00024-X,
  3. Bonet J, Lok TS (1999) Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. Comput Methods Appl Mech Eng 180(1–2): 97–115, doi: 10.1016/S0045-7825(99)00051-1,
  4. Chatelain P, Backaert S, Winckelmans G, Kern S (2013) Large eddy simulation of wind turbine wakes. Flow Turbul Combust 91(3):587–605. doi: 10.1007/s10494-013-9474-8 CrossRefGoogle Scholar
  5. Chertock A, Levy D (2001) Particle methods for dispersive equations. J Comput Phys 171(2): 708–730, doi: 10.1006/jcph.2001.6803,
  6. Chertock A, Levy D (2002) A particle method for the KdV equation. J Sci Comput 17(1—-4):491–499. doi: 10.1023/A:1015106210404 MathSciNetCrossRefMATHGoogle Scholar
  7. Chorin AJ (1973) Numerical study of slightly viscous flow. J Fluid Mech 57(4):785–796. doi: 10.1017/S0022112073002016 MathSciNetCrossRefGoogle Scholar
  8. Christiansen J (1997) Numerical simulation of hydrodynamics by the method of point vortices. J Comput Phys 135(2):189–197, doi: 10.1006/jcph.1997.5701,
  9. Colagrossi A, Landrini M (2003) Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J Comput Phys 191(2): 448–475, doi: 10.1016/S0021-9991(03)00324-3,
  10. Cortez R (1997) Convergence of high-order deterministic particle methods for the convection-diffusion equation. Commun Pure Appl Math 50(L):1235–1260MathSciNetCrossRefMATHGoogle Scholar
  11. Cottet GH, Poncet P (2004) Advances in direct numerical simulations of 3d wall-bounded flows by vortex-in-cell methods. J Comput Phys 193(1):136–158, doi: 10.1016/,
  12. Degond P, Mas-Gallic S (1989) The weighted particle method for convection-diffusion equations. Part I: The case of an isotropic viscosity. Math Comput 53(188):485–507. doi: 10.2307/2008716 MathSciNetMATHGoogle Scholar
  13. Degond P, Mustieles FJ (1990) A deterministic approximation of diffusion equations using particles. SIAM J Sci Stat Comput 11(2): 293–310, doi: 10.1137/0911018,
  14. Dynnikov Y, Dynnikova G (2011) Numerical stability and numerical viscosity in certain meshless vortex methods as applied to the navier-stokes and heat equations. Comput Math Math Phys 51:1792–1804. doi: 10.1134/S096554251110006X MathSciNetCrossRefMATHGoogle Scholar
  15. Dynnikova G (2011) Calculation of three-dimensional flows of an incompressible fluid based on a dipole representation of vorticity. Dokl Phys 56(3):163–166. doi: 10.1134/S1028335811030025 CrossRefGoogle Scholar
  16. Eldredge JD, Leonard A, Colonius T (2002) A general deterministic treatment of derivatives in particle methods. J Comput Phys 180(2): 686–709. doi: 10.1006/jcph.2002.7112,
  17. Fronteau J, Combis P (1984) A lie admissible method of integration of folkler-plank equations with non linear coefficients (exact and numerical solutions). Hadronic J 7:911–930MathSciNetMATHGoogle Scholar
  18. Gambino G, Lombardo M, Sammartino M (2009) A velocity-diffusion method for a lotka-volterra system with nonlinear cross and self-diffusion. Appl Numer Math 59(5):1059–1074. doi: 10.1016/j.apnum.2008.05.002,
  19. Grant J, Marshall J (2005) Diffusion velocity for a three-dimensional vorticity field. Theor Comput Fluid Dyn 19:377–390. doi: 10.1007/s00162-005-0004-8 CrossRefMATHGoogle Scholar
  20. Guvernyuk S, Dynnikova G (2007) Modeling the flow past an oscillating airfoil by the method of viscous vortex domains. Fluid Dyn 42:1–11. doi: 10.1134/S0015462807010012 MathSciNetCrossRefMATHGoogle Scholar
  21. Kempka S, Strickland J (1993) A method to simulate viscous diffusion of vorticity by convective transport of vortices at a non-solenoidal velocity. Technical Report SAND-93-1763, Sandia Laboratory,
  22. Lacombe G (1999) Analyse d’une équation de vitesse de diffusion. Comptes Rendus de l’Académie des Sciences—Series I—Mathematics, vol 329, no 5, pp 383–386. doi: 10.1016/S0764-4442(00)88610-3, URL
  23. Lacombe G, Mas-Gallic S (1999) Presentation and analysis of a diffusion-velocity method. ESAIM Proc 7:225–233. doi: 10.1051/proc:1999021 MathSciNetCrossRefMATHGoogle Scholar
  24. Leonard A (1980) Vortex methods for flow simulation. J Comput Phys 37(3): 289–335. doi: 10.1016/0021-9991(80)90040-6,
  25. Lions PL, Mas-Gallic S (2001) Une méthode particulaire déterministe pour des équations diffusives non linéaires. Comptes Rendus de l’Académie des Sciences—Series I—Mathematics 332(4), pp 369–376. doi: 10.1016/S0764-4442(00)01795-X,
  26. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8—-9):1081–1106. doi: 10.1002/fld.1650200824 MathSciNetCrossRefMATHGoogle Scholar
  27. Liu WK, Chen Y, Uras R, Chang CT (1996) Generalized multiple scale reproducing kernel particle methods. Comput Methods Appl Mech Eng 139(1–4): 91–157. doi: 10.1016/S0045-7825(96)01081-X,
  28. Mansfield JR (1997) A dynamic Lagrangian Large Eddy Simulation scheme for the vorticity transport equation. PhD thesis, John Hopkins University, Baltimore, Maryland, USAGoogle Scholar
  29. Mansfield JR, Knio OM, Meneveau C (1996) Towards lagrangian large vortex simulation. ESAIM Proc 1:49–64. doi: 10.1051/proc:1996019 MathSciNetCrossRefMATHGoogle Scholar
  30. Mansfield JR, Knio OM, Meneveau C (1998) A dynamic LES scheme for the vorticity transport equation: Formulation and a priori tests. J Comput Phys 145(2): 693–730. doi: 10.1006/jcph.1998.6051,
  31. Mas-Gallic S (1999) A presentation of the diffusion velocity method. In: Leach P, Bouquet S, Rouet JL, Fijalkow E (eds) Dynamical Systems, Plasmas and Gravitation, Lecture Notes in Physics. Springer, Berlin, pp 74–81. doi: 10.1007/BFb0105914 CrossRefGoogle Scholar
  32. Meneveau C, Katz J (2000) Scale-invariance and turbulence models for large-eddy simulation. Ann Rev Fluid Mech 32(1): 1–32. doi: 10.1146/annurev.fluid.32.1.1,
  33. Milane R, Nourazar S (1995) On the turbulent diffusion velocity in mixing layer simulated using the vortex method and the subgrid scale vorticity model. Mech Res Commun 22(4): 327–333. doi: 10.1016/0093-6413(95)00032-M,
  34. Milane R, Nourazar S (1997) Large-eddy simulation of mixing layer using vortex method: Effect of subgrid-scale models on early development. Mech Res Commun 24(2): 215–221. doi: 10.1016/S0093-6413(97)00015-3,
  35. Milane RE (2004) Large eddy simulation (2d) using diffusion-velocity method and vortex-in-cell. Int J Numer Methods Fluids 44(8): 837–860. doi: 10.1002/fld.673
  36. Mustieles FJ (1990) L’équation de Boltzmann des semiconducteurs. Étude mathématique et simulation numérique. PhD thesis, école PolytechniqueGoogle Scholar
  37. Mycek P, Pinon G, Germain G, Rivoalen E (2013) A self-regularising DVM-PSE method for the modelling of diffusion in particle methods. Comptes Rendus Mécanique 341(9–10):709–714. doi: 10.1016/j.crme.2013.08.002,
  38. Ogami Y (1999) Simulation of heat-vortex interaction by the diffusion velocity method. ESAIM Proc 7:313–324. doi: 10.1051/proc:1999029
  39. Ogami Y, Akamatsu T (1991) Viscous flow simulation using the discrete vortex model-the diffusion velocity method. Comput Fluids 19(3–4): 433–441. doi: 10.1016/0045-7930(91)90068-S,
  40. Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved sph method: Towards higher order convergence. J Comput Phys 225(2): 1472–1492. doi: 10.1016/,
  41. Pinon G, Mycek P, Germain G, Rivoalen E (2012) Numerical simulation of the wake of marine current turbines with a particle method. Renew Energy 46(0): 111–126. doi: 10.1016/j.renene.2012.03.037,
  42. Rivoalen E, Huberson S (1999) Numerical simulation of axisymmetric viscous flows by means of a particle method. J Comput Phys 152(1): 1–31. doi: 10.1006/jcph.1999.6210,
  43. Rivoalen E, Huberson S, Hauville F (1997) Simulation numérique des équations de Navier-Stokes 3D par une méthode particulaire. Comptes Rendus de l’Académie des Sciences—Series IIB—Mechanics-Physics-Chemistry-Astronomy, vol 324, no 9, pp. 543–549. doi: 10.1016/S1251-8069(97)83187-9,
  44. Sagaut P (2006) Large Eddy Simulation for Incompressible Flows: an Introduction. Scientific Computation, Springer.
  45. Schrader B, Reboux S, Sbalzarini IF (2010) Discretization correction of general integral pse operators for particle methods. J Comput Phys 229(11): 4159–4182. doi: 10.1016/,
  46. Strickland JH, Kempka SN, Wolfe WP (1996) Viscous diffusion of vorticity using the diffusion velocity concept. ESAIM Proc 1:135–151. doi: 10.1051/proc:1996033 MathSciNetCrossRefMATHGoogle Scholar
  47. Winckelmans G, Cocle R, Dufresne L, Capart R (2005) Vortex methods and their application to trailing wake vortex simulations. Comptes Rendus Physique 6(4–5):467–486. doi: 10.1016/j.crhy.2005.05.001,

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014

Authors and Affiliations

  • Paul Mycek
    • 1
  • 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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