Integrodifferential equations for simulating moving dislocations

In this talk, we introduce an integrodifferential equation motivated by dislocation dynamics in material sciences [3]. This equation involves a stiff convolution kernel that is nonlocal in time and in space (there is a memory effect). Its simulation raises two types of questions: the stability of the employed numerical method and the algorithmic efficiency of the method, both from the points of view of the complexity and the memory requirement. The challenge is to store as little memory as possible, and to use it. We describe two families of fast methods dealing improving the algorithmic efficiency: the first one relying on the Fast Fourier Transform [1], and the second on the Laplace transform [2].

1. E. Hairer, C. Lubich, and M. Schlichte. Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Statist. Comput., 6(3):532--541, 1985.
2. C. Lubich and A. Schädle. Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput., 24(1):161--182, 2002.
3. P. Pellegrini. Dynamic Peierls-Nabarro equations for elastically isotropic crystals. Phys. Rev. B, 81:024101, 2010