耦合非线性Schrouml;dinger方程的紧致差分格式任务书

 2021-08-20 12:08

1. 毕业设计（论文）主要目标：

耦合非线性Schrdinger方程在物理学的众多领域都有非常广泛的应用，例如非线性光学、凝聚态物理、等离子体物理等。对于该方程组的求解一直以来都是物理学中国际热点和难点问题。然而该方程在绝大多数定解条件下都很难得到精确解显示表达式，因而数值求解就显得尤为重要。在本文中，我们致力于对耦合非线性Schrdinger方程提出一种新的紧致有限差分格式。引入一个新的能量泛函，证明新格式在离散意义下保持总质量和总能量守恒。运用Taylor展开，给出格式的局部截断误差。通过数值算例验证格式的精度，并对一些物理想象进行模拟。由于本课题解决的问题一直以来都是物理学中当前的国际热点问题，构造对耦合非线性Schrdinger方程的紧致差分格式就显得尤为重要。

2. 毕业设计（论文）主要内容：

1. 对要研究的耦合非线性方程组的物理背景和已有数学、数值研究进行综述；2. 运用紧致有限差分法对耦合非线性Schrdinger方程组提出了一个线性化、解耦的有限差分格式；3. 引入一类新型能量泛函，证明新格式在离散意义上保持总质量和总能量守恒；4.运用Taylor展开，详细推导出格式的局部截断误差5. 通过数值结果来验证格式的精度和并模拟两个孤立波的碰撞现象；6.对我们的主要工作进行简明的总结。

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