Fourier-based numerical approximation of the Weertman equation for moving dislocations
This work provides an efficient numerical method for solving a key equation in materials science, but the improvement is incremental over existing approaches.
The authors developed a Fourier-based preconditioned collocation scheme to numerically solve the Weertman equation for moving dislocations, achieving robust convergence with large time steps. The method efficiently handles various nonlinearities, as demonstrated by numerical results.
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.