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

## TL;DR

This paper presents a Fourier-based numerical method for approximating solutions to the Weertman equation, modeling moving dislocations, using a time-dependent formulation and a preconditioned collocation scheme for efficiency.

## Contribution

It introduces a novel Fourier transform-based collocation scheme with large time steps for solving the nonlinear Weertman equation.

## Key findings

- Efficient numerical approximation of the Weertman equation.
- Robust method capturing solutions with large time steps.
- Applicable to various nonlinearities in dislocation modeling.

## 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.