Title: Fourier-based numerical approximation of the Weertman equation for moving dislocations,
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.