基于傅里叶变换的Weertman方程数值近似
Fourier-based numerical approximation of the Weertman equation for moving dislocations
AI总结：
本文提出了一种基于傅里叶变换的预条件置bel方案，用于求解Weertman方程的非线性反应-对流-扩散方程，展示了该方法在处理不同非线性情况下的高效性。
摘要：
本文讨论了非线性反应-对流-扩散方程的数值近似，该方程是Weertman方程的无量纲形式，用于建模材料科学中稳态移动的位错。当对流项设为零时，该方程退化为著名的Peierls-Nabarro方程。该方法基于时间依赖性的公式，允许将研究方程视为其长时间极限。引入基于傅里叶变换的预条件置bel方案，所提出的迭代数值方法解决了时间依赖性问题，在收敛时提供所需的Weertman方程数值解。尽管该方法基于显式时间演化方案，但允许使用大时间步长，并以稳健的方式捕捉解。数值结果展示了该方法在多种非线性情况下的有效性。
英文摘要：
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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