User menu

A comparison of three Eulerian numerical methods for fractional-order transport models

Bibliographic reference Hanert, Emmanuel. A comparison of three Eulerian numerical methods for fractional-order transport models. In: Environmental Fluid Mechanics (Dordrecht, 2001), Vol. 10, no. 1-2, p. 7-20 (2010)
Permanent URL
  1. Batchelor GK (1950) The application of the similarity theory of turbulence to atmospheric diffusion. Quart J R Meteorol Soc 76: 133–146
  2. Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection-dispersion equation. Water Resour Res 36: 1403–1412
  3. Benson DA, Wheatcraft SW, Meerschaert MM (2000) The fractional-order governing equation of levy motion. Water Resour Res 36: 1413–1423
  4. Berkowicz R, Prahm L (1979) Generalization of K theory for turbulent diffusion. Part 1: spectral diffusivity concept. J Appl Meteorol 18: 266–272
  5. Berkowitz B, Cortis A, Dentz M, Scher H (2006) Modelling non-Fickian transport in geological formations as a continuous time random walk. Rev Geophys 44(RG2003): 3
  6. Boyd JP (2001) Chebyshev and Fourier spectral methods, 2nd edn. Dover Publications, New York
  7. Chaves AS (1998) A fractional diffusion equation to describe Lévy flights. Phys Lett A 239: 13–16
  8. Cushman-Roisin B (2008) Beyond eddy diffusivity: an alternative model for turbulent dispersion. Environ Fluid Mech 8: 543–549
  9. Cushman-Roisin B, Jenkins AD (2006) On a non-local parameterization for shear turbulence and the uniqueness of its solutions. Boundary-Layer Meteorol 118: 69–82
  10. Davies RE (1983) Oceanic property transport, Lagrangian particle statistics, and their prediction. J Mar Res 41: 163–194
  11. Deng ZQ, Bengtson L, Singh VP (2006) Parameter estimation for fractional dispersion model for rivers. Environ Fluid Mech 6: 451–475
  12. Durbin PA (1980) A stochastic model of two-particle dispersion and concentration fluctuations in homogeneous turbulence. J Fluid Mech 100: 279–302
  13. Einstein A (1905) Uber die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik 17: 549–560
  14. Feller W (1971) An introduction to probability theory and its applications, vol II. Wiley, New York
  15. Fix GJ, Roop JP (2004) Least square finite-element solution of a fractional order two-point boundary value problem. Comput Math Appl 48: 1017–1033
  16. Frippiat CC, Holeyman AE (2008) A comparative review of upscaling methods for solute transport in heterogenenous porous media. J Hydrol 362: 150–176
  17. Gnedenko B, Kolmogorov A (1954) Limit distributions for sums of independent random variables. Addison-Wesley, Cambridge, MA
  18. Huang G, Huang Q, Zhan H (2006) Evidence of one-dimensional scale-dependent fractional advection-dispersion. J Contam Hydrol 85: 53–71
  19. Jenkins AD (1985) Simulation of turbulent dispersion using a simple random model of the flow field. Appl Math Model 9: 239–245
  20. Kim S, Kavvas ML (2006) Generalized Fick’s law and fractional ADE for pollution transport in a river: detailed derivation. J Hydrol Eng 11(1): 80–83
  21. Lévy P (1954) Théorie de l’Addition des Variables Aléatoires. Gauthier-Villars, Paris
  22. Meerschaert MM, Benson DA, Bäumer B (1999) Multidimensional advection and fractional dispersion. Phys Lett A 59: 5026–5028
  23. Meerschaert MM, Tadjeran C (2004) Finite difference approximations for fractional advection-diffusion flow equations. J Comput Appl Math 172: 65–77
  24. Okubo A (1971) Oceanic diffusion diagrams. Deep Sea Res 18: 789–802
  25. Pachepsky Y, Timlin D, Rawls W (2003) Generalized Richards’ equation to simulate water transport in unsaturated soils. J Hydrol 272: 3–13
  26. Podlubny I (1999) Fractional differential equations: mathematics in science and engineering, vol 198. Academic Press, New York
  27. Richardson LF (1926) Atmospheric diffusion shown on a distance-neighbour graph. Proc R Soc Lond 110: 709–737
  28. Richardson Lewis F., Stommel Henry, NOTE ON EDDY DIFFUSION IN THE SEA, 10.1175/1520-0469(1948)005<0238:noedit>;2
  29. Roop JP (2006) Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in $${\mathbb{R}^2}$$ . J Comput Appl Math 193: 243–268
  30. Schumer R, Benson DA, Meerschaert MM, Wheatcraft JW (2001) Eulerian derivation of the fractional advection-dispersion equation. J Contam Hydrol 48: 69–88
  31. Stommel H (1949) Horizontal diffusion due to oceanic turbulence. J Mar Res 8: 199–225
  32. Tadjeran C, Meerschaert MM, Scheffler H-P (2006) A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 213: 205–213
  33. Zhang Y, Benson DA, Reeves DM (2009) Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Adv Water Resour 32(4): 561–581