非局部Sine-Gordon方程色散解的一种谱方法
A spectral method for dispersive solutions of the nonlocal Sine-Gordon equation
AI总结：
本文针对近场动力学Sine-Gordon方程的非局部孤子解，提出切比雪夫谱方法结合Stormer-Verlet格式的数值方案，通过数值实验验证了其捕捉色散效应和分析内能行为的能力。
摘要：
出于对分析近场动力学材料所需的严格且可靠数值工具的需求，作者提出一个能够捕捉由Sine-Gordon方程的近场动力学公式所得到的非局部类孤子解的色散特征的模型。本文通过切比雪夫多项式节点上的谱方法结合用于时间演化的Stormer-Verlet格式，对具有局部Neumann边界条件的近场动力学Sine-Gordon方程所关联的Cauchy问题进行分析。选择使用谱方法的原因在于其可达到的数值精度，而切比雪夫多项式确实允许直接实现局部边界条件。文中提出了若干数值实验以全面描述该格式的能力。具体而言，展示了特定近场动力学核的色散效应，同时研究了所指定近场动力学算子的内能行为。
英文摘要：
Moved by the need for rigorous and reliable numerical tools for the analysis of peridynamic materials, the authors propose a model able to capture the dispersive features of nonlocal soliton-like solutions obtained by a peridynamic formulation of the Sine-Gordon equation. The analysis of the Cauchy problem associated to the peridynamic Sine-Gordon equation with local Neumann boundary condition is performed in this work through a spectral method on Chebyshev polynomials nodes joined with the Stormer-Verlet scheme for the time evolution. The choice for using the spectral method resides in the resulting reachable numerical accuracy, while, indeed, Chebyshev polynomials allow straightforward implementation of local boundary conditions. Several numerical experiments are proposed for thoroughly describe the ability of such scheme. Specifically, dispersive effects of the specific peridynamic kernel are demonstrated, while the internal energy behavior of the specified peridynamic operator is studied.
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