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On the spectral vanishing viscosity method for periodic fractional conservation laws

Authors: Simone Cifani and Espen R. Jakobsen
Journal: Math. Comp. 82 (2013), 1489-1514
MSC (2010): Primary 65M70, 35K59, 35R09; Secondary 65M15, 65M12, 35K57, 35R11
Published electronically: March 19, 2013
MathSciNet review: 3042572
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Abstract: We introduce and analyze a spectral vanishing viscosity approximation of periodic fractional conservation laws. The fractional part of these equations can be a fractional Laplacian or other non-local operators that are generators of pure jump Lévy processes. To accommodate for shock solutions, we first extend to the periodic setting the Kružkov-Alibaud entropy formulation and prove well-posedness. Then we introduce the numerical method, which is a non-linear Fourier Galerkin method with an additional spectral viscosity term. This type of approximation was first introduced by Tadmor for pure conservation laws. We prove that this non-monotone method converges to the entropy solution of the problem, that it retains the spectral accuracy of the Fourier method, and that it diagonalizes the fractional term reducing dramatically the computational cost induced by this term. We also derive a robust $ L^1$-error estimate, and provide numerical experiments for the fractional Burgers' equation.

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Additional Information

Simone Cifani
Affiliation: Department of Mathematics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway

Espen R. Jakobsen
Affiliation: Department of Mathematics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway

Keywords: Fractional/fractal conservation laws, entropy solutions, Fourier spectral methods, spectral vanishing viscosity, convergence, error estimate.
Received by editor(s): November 15, 2010
Received by editor(s) in revised form: November 29, 2011
Published electronically: March 19, 2013
Additional Notes: This research was supported by the Research Council of Norway (NFR) through the project “Integro-PDEs: Numerical Methods, Analysis, and Applications to Finance”.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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