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Well-posedness and numerical analysis of a one-dimensional non-local transport equation modelling dislocations dynamics


Authors: A. Ghorbel and R. Monneau
Journal: Math. Comp. 79 (2010), 1535-1564
MSC (2010): Primary 35F20, 35F25, 35K55, 49L25, 65N06, 65N12, 74N05
DOI: https://doi.org/10.1090/S0025-5718-10-02326-4
Published electronically: March 23, 2010
MathSciNet review: 2630002
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Abstract: We consider a situation where dislocations are parallel lines moving in a single plane. For this simple geometry, dislocations dynamics is modeled by a one-dimensional non-local transport equation. We prove a result of existence and uniqueness for all time of the continuous viscosity solution for this equation. A finite difference scheme is proposed to approximate the continuous viscosity solution. We also prove an error estimate result between the continuous solution and the discrete solution, and we provide some simulations.


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A. Ghorbel
Affiliation: CERMICS, École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cédex 2, France
Email: ghorbel@cermics.enpc.fr

R. Monneau
Affiliation: CERMICS, École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cédex 2, France
Email: monneau@cermics.enpc.fr

DOI: https://doi.org/10.1090/S0025-5718-10-02326-4
Keywords: Dislocations dynamics, Peach-Koehler force, transport equation, eikonal equation, Hamilton-Jacobi equation, non-local equation, continuous viscosity solutions, convergence of numerical scheme, finite difference scheme.
Received by editor(s): January 13, 2009
Published electronically: March 23, 2010
Article copyright: © Copyright 2010 American Mathematical Society

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