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Mathematics of Computation

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A numerical method for fractal conservation laws

Author: Jérôme Droniou
Journal: Math. Comp. 79 (2010), 95-124
MSC (2000): Primary 65M12, 35L65, 35S10, 45K05
Published electronically: July 29, 2009
MathSciNet review: 2552219
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Abstract: We consider a fractal scalar conservation law, that is to say, a conservation law modified by a fractional power of the Laplace operator, and we propose a numerical method to approximate its solutions. We make a theoretical study of the method, proving in the case of an initial data belonging to $ L^\infty\cap BV$ that the approximate solutions converge in $ L^\infty$ weak-$ *$ and in $ L^p$ strong for $ p<\infty$, and we give numerical results showing the efficiency of the scheme and illustrating qualitative properties of the solution to the fractal conservation law.

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Additional Information

Jérôme Droniou
Affiliation: Université Montpellier 2, Institut de Mathématiques et de Modélisation de Montpellier, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France

Keywords: Conservation laws, L\'evy operator, fractal operator, integral operator, numerical scheme, proof of convergence, numerical results.
Received by editor(s): April 25, 2009
Received by editor(s) in revised form: March 23, 2009
Published electronically: July 29, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.