Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Splitting methods for the nonlocal Fowler equation

Authors: Afaf Bouharguane and Rémi Carles
Journal: Math. Comp. 83 (2014), 1121-1141
MSC (2010): Primary 65M15; Secondary 35K59, 86A05
Published electronically: August 5, 2013
MathSciNet review: 3167452
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlocal scalar conservation law proposed by Andrew C. Fowler to describe the dynamics of dunes, and we develop a numerical procedure based on splitting methods to approximate its solutions. We begin by proving the convergence of the well-known Lie formula, which is an approximation of the exact solution of order one in time. We next use the split-step Fourier method to approximate the continuous problem using the fast Fourier transform and the finite difference method. Our numerical experiments confirm the theoretical results.

References [Enhancements On Off] (What's this?)

  • [1] Nathaël Alibaud, Pascal Azerad, and Damien Isèbe, A non-monotone nonlocal conservation law for dune morphodynamics, Differential Integral Equations 23 (2010), no. 1-2, 155-188. MR 2588807 (2011b:35310)
  • [2] Borys Alvarez-Samaniego and Pascal Azerad, Existence of travelling-wave solutions and local well-posedness of the Fowler equation, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 4, 671-692. MR 2552069 (2010i:35195),
  • [3] P. Azerad and A. Bouharguane, Finite difference approximations for a fractional diffusion/anti-diffusion equation, preprint,, 2011.
  • [4] Pascal Azerad, Afaf Bouharguane, and Jean-François Crouzet, Simultaneous denoising and enhancement of signals by a fractal conservation law, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 2, 867-881. MR 2834457 (2012i:94061),
  • [5] Christophe Besse, Brigitte Bidégaray, and Stéphane Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 40 (2002), no. 1, 26-40 (electronic). MR 1921908 (2003k:65099),
  • [6] Afaf Bouharguane, On the instability of a nonlocal conservation law, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 3, 419-426. MR 2861816 (2012k:35263),
  • [7] S. Descombes and M. Thalhammer, The Lie-Trotter splitting for nonlinear evolutionary problems with critical parameters: A compact local error representation and application to nonlinear Schrödinger equations in the semiclassical regime, IMA J. Numer. Anal. 33 (2013), 722-745, DOI 10.1093/imanum/drs021. MR 3047949
  • [8] Kai Diethelm and Neville J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput. 154 (2004), no. 3, 621-640. MR 2072809 (2005h:34006),
  • [9] J. Droniou and C. Imbert, Solutions de viscosité et solutions variationnelles pour EDP non-linéaires, Lecture notes, available at, 2004.
  • [10] Erwan Faou, Analysis of splitting methods for reaction-diffusion problems using stochastic calculus, Math. Comp. 78 (2009), no. 267, 1467-1483. MR 2501058 (2010k:65157),
  • [11] A. C. Fowler, Mathematics and the environment, Lecture notes, available at
  • [12] -, Dunes and drumlins, Geomorphological fluid mechanics (A. Provenzale and N. Balmforth, eds.), vol. 211, Springer-Verlag, Berlin, 2001, pp. 430-454.
  • [13] Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie, and Nils Henrik Risebro, Splitting methods for partial differential equations with rough solutions, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2010. Analysis and MATLAB programs. MR 2662342 (2011j:65002)
  • [14] Helge Holden, Kenneth H. Karlsen, Nils Henrik Risebro, and Terence Tao, Operator splitting for the KdV equation, Math. Comp. 80 (2011), no. 274, 821-846. MR 2772097 (2012d:35321),
  • [15] Helge Holden, Christian Lubich, and Nils Henrik Risebro, Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp. 82 (2013), no. 281, 173-185. MR 2983020,
  • [16] Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. MR 951744 (90f:35162),
  • [17] Christian Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp. 77 (2008), no. 264, 2141-2153. MR 2429878 (2009d:65114),
  • [18] David L. Ropp and John N. Shadid, Stability of operator splitting methods for systems with indefinite operators: advection-diffusion-reaction systems, J. Comput. Phys. 228 (2009), no. 9, 3508-3516. MR 2517530 (2010e:65141),
  • [19] Andrea Sacchetti, Spectral splitting method for nonlinear Schrödinger equations with singular potential, J. Comput. Phys. 227 (2007), no. 2, 1483-1499. MR 2442402 (2009j:65268),
  • [20] Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506-517. MR 0235754 (38 #4057)
  • [21] Thiab R. Taha and Mark J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys. 55 (1984), no. 2, 203-230. MR 762363 (86e:65128b),
  • [22] Haiping Ye, Jianming Gao, and Yongsheng Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075-1081. MR 2290034,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M15, 35K59, 86A05

Retrieve articles in all journals with MSC (2010): 65M15, 35K59, 86A05

Additional Information

Afaf Bouharguane
Affiliation: Institut de Mathématiques de Bordeaux, Univ. Bordeaux 1, 33405 Talence cedex, France

Rémi Carles
Affiliation: CNRS & Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France

Keywords: Nonlocal operator, numerical time integration, operator splitting, split-step Fourier method, stability, error analysis
Received by editor(s): September 4, 2011
Received by editor(s) in revised form: March 21, 2012, June 11, 2012, August 1, 2012, and September 17, 2012
Published electronically: August 5, 2013
Additional Notes: This work was supported by the French ANR project MATHOCEAN, ANR-08-BLAN-0301-02.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society