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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
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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.


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Additional Information

Afaf Bouharguane
Affiliation: Institut de Mathématiques de Bordeaux, Univ. Bordeaux 1, 33405 Talence cedex, France
Email: Afaf.Bouharguane@math.u-bordeaux1.fr

Rémi Carles
Affiliation: CNRS & Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France
Email: Remi.Carles@math.cnrs.fr

DOI: https://doi.org/10.1090/S0025-5718-2013-02757-3
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.