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arXiv · 1704.04489

Fourier-based numerical approximation of the Weertman equation for moving dislocations

Marc Josien·Yves-Patrick Pellegrini·Frédéric Legoll·Claude Le Bris
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Abstract

This work discusses the numerical approximation of a nonlinear reaction-advection-diffusion equation, which is a dimensionless form of the Weertman equation. This equation models steadily-moving dislocations in materials science. It reduces to the celebrated Peierls-Nabarro equation when its advection term is set to zero. The approach rests on considering a time-dependent formulation, which admits the equation under study as its long-time limit. Introducing a Preconditioned Collocation Scheme based on Fourier transforms, the iterative numerical method presented solves the time-dependent problem, delivering at convergence the desired numerical solution to the Weertman equation. Although it rests on an explicit time-evolution scheme, the method allows for large time steps, and captures the solution in a robust manner. Numerical results illustrate the efficiency of the approach for several types of nonlinearities.

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Marc Josien, Yves-Patrick Pellegrini, Frédéric Legoll, Claude Le Bris. 2017-04-14. Fourier-based numerical approximation of the Weertman equation for moving dislocations. https://doi.org/10.1002/nme.5723

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