We consider a two dimensional particle diffusing in the presence of a fast cellular flow confined to a finite domain. If the flow amplitude $A$ is held fixed and the number of cells $L^2\rightarrow\infty$, then the problem homogenizes; this has been well studied. Also well studied is the limit when $L$ is fixed and $A\rightarrow\infty$. In this case the solution averages along stream lines. The double limit as both the flow amplitude $A\rightarrow\infty$ and the number of cells $L^2\rightarrow\infty$ was recently studied [G. Iyer et al., preprint, arXiv:1108.0074]; one observes a sharp transition between the homogenization and averaging regimes occurring at $A\approx L^4$. This paper numerically studies a few theoretically unresolved aspects of this problem when both $A$ and $L$ are large that were left open in [G. Iyer et al., preprint, arXiv:1108.0074] using the numerical method devised in [G. A. Pavliotis, A. M. Stewart, and K. C. Zygalakis, J. Comput. Phys., 228 (2009), pp. 1030--1055]. Our treatment of the numerical method uses recent developments in the theory of modified equations for numerical integrators of stochastic differential equations [K. C. Zygalakis, SIAM J. Sci. Comput., 33 (2001), pp. 102--130].


  1. homogenization
  2. averaging
  3. backward error analysis
  4. Monte Carlo methods

MSC codes

  1. 35B27
  2. 70K65
  3. 37M15
  4. 60H35

Get full access to this article

View all available purchase options and get full access to this article.


A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North--Holland, Amsterdam, 1978.
H. Berestycki, F. Hamel, and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), pp. 451--480.
P. Castiglione and A. Crisanti, Dispersion of passive tracers in a velocity field with non-$\delta$-correlated noise, Phys. Rev. E (3), 59 (1999), pp. 3926--3934.
P. Chartier, E. Faou, and A. Murua, An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants, Numer. Math., 103 (2006), pp. 575--590.
S. Childress, Alpha-effect in flux ropes and sheets, Phys. Earth Planet Inter., 20 (1979), pp. 172--180.
G. T. Csanady, ed., Turbulent Diffusion in the Environment, Vol. 3, D. Reidel, Dordrecht, The Netherlands, 1973.
J. Douglas, Jr., and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), pp. 871--885.
G. Falkovich, A. Fouxon, and M. G. Stepanov, Acceleration of rain initiation by cloud turbulence, Nature, 419 (2002), pp. 151--154.
G. Falkovich, K. Gawedzki, and M. Vergassola, Particles and fields in fluid turbulence, Rev. Modern Phys., 73 (2001), pp. 913--975.
A. Fannjiang, A. Kiselev, and L. Ryzhik, Quenching of reaction by cellular flows, Geom. Funct. Anal., 16 (2006), pp. 40--69.
A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), pp. 333--408.
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 2nd ed., Grundlehren Math. Wisse. 260, Springer-Verlag, New York, 1998.
Y. Gorb, D. Nam, and A. Novikov, Numerical simulations of diffusion in cellular flows at high Péclet numbers, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), pp. 75--92.
E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer Ser. Comput. Math. 31, Springer-Verlag, Berlin, 2006.
S. Heinze, Diffusion-advection in cellular flows with large Peclet numbers, Arch. Ration. Mech. Anal., 168 (2003), pp. 329--342.
G. Iyer, T. Komorowski, A. Novikov, and L. Ryzhik, From homogenization to averaging in cellular flows, preprint, arXiv:1108.0074.
V. V. Jikov, S. M. Kozlov, and O. A. Ole\u\inik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
R. B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp., 32 (1978), pp. 1025--1039.
A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), pp. 309--358.
D. L. Koch, R. G. Cox, H. Brenner, and J. F. Brady, The effect of order on dispersion in porous media, J. Fluid Mech., 200 (1989), pp. 173--188.
L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fields, 129 (2004), pp. 37--62.
A. J. Majda and P. R. Krammer, Simplified models for turbulent diffusion: Theory, numerical modeling, and physical phenomena, Phys. Rep., 314 (1999), pp. 237--574.
J. J. H. Miller, E. O'Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, River Edge, NJ, 1996.
A. Novikov, G. Papanicolaou, and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math., 58 (2005), pp. 867--922.
B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th ed., Universitext, Springer-Verlag, Berlin, 2003.
G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Texts Appl. Math. 53, Springer, New York, 2008.
G. A. Pavliotis and A. M. Stuart, Periodic homogenization for inertial particles, Phys. D, 204 (2005), pp. 161--187.
G. A. Pavliotis, A. M. Stuart, and K. C. Zygalakis, Calculating effective diffusiveness in the limit of vanishing molecular diffusion, J. Comput. Phys., 228 (2009), pp. 1030--1055.
G. R. W. Quispel and D. I. McLaren, Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows, J. Comput. Phys., 186 (2003), pp. 308--316.
P. B. Rhines and W. R. Young, How rapidly is passive scalar mixed within closed streamlines?, J. Fluid Mech., 133 (1983), pp. 135--145.
A. Rößler, Stochastic Taylor expansions for the expectation of functionals of diffusion processes, Stoch. Anal. Appl., 22 (2004), pp. 1553--1576.
T. Shardlow, Modified equations for stochastic differential equations, BIT, 46 (2006), pp. 111--125.
R. A. Shaw, Particle-turbulence interactions in atmosphere clouds, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech. 35, Annual Reviews, Palo Alto, CA, 2003, pp. 183--227.
W. Young, A. Pumir, and Y. Pomeau, Anomalous diffusion of tracer in convection rolls, Phys. Fluids A, 1 (1989), pp. 462--469.
A. Zlatoš, Reaction-diffusion front speed enhancement by flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), pp. 711--726.
K. C. Zygalakis, On the existence and the applications of modified equations for stochastic differential equations, SIAM J. Sci. Comput., 33 (2011), pp. 102--130.

Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1046 - 1058
ISSN (online): 1540-3467


Submitted: 4 January 2012
Accepted: 4 June 2012
Published online: 13 September 2012


  1. homogenization
  2. averaging
  3. backward error analysis
  4. Monte Carlo methods

MSC codes

  1. 35B27
  2. 70K65
  3. 37M15
  4. 60H35



Konstantinos C. Zygalakis

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media