Methods for Solving Nonlinear Differential Equations Describing Water Waves

在这项工作中，我们探讨了两个主题：计算描述水波的方程的解和分析这些解的稳定性。研究项目的重点是冰下的波浪。随着最近气候变化和全球变暖的趋势导致海冰融化，这对加拿大北部社区的物资运输产生了很大影响。这项工作的长期目标是有助于理解冰-水相互作用。我们特别希望了解在什么条件下冰会破裂，例如，如果用卡车在结冰的道路上运输物资。该方法将被分解为短期目标。首先，我们将使用高性能计算技术来加速现有的计算冰下波浪的代码。使用更有效的代码，我们将调用一个更物理的模型来描述相互作用，包括冰的不均匀性以及能量的耗散。为了检验结果方程的物理解是怎样的，我们将修改现有的稳定性分析方法，使其也包括耗散效应。由于在非线性状态下会出现更多的极端波，我们还将扩展稳定性分析以捕获这些类型的解。这项工作的影响不仅有助于理解冰下的波浪，而且所开发的方法将适用于广泛的学科，无论何时通过偏微分方程对物理现象进行建模。

In this work we explore two themes: computing solutions to equations describing water waves and analysing the stability of these solutions. The focus of the research program is on waves under ice. With a recent trend of climate change and global warming resulting in melting of sea ice, this has a large impact on transportation of supplies to communities in northern Canada. The long term vision of this work aims to contribute to understanding ice-water interactions. In particular we hope to understand under which conditions the ice can break, for example, if supplies are transported in a truck travelling over an ice road. The approach will be broken down into shorter term goals. First, we will use high performance computing techniques to speed up existing code for computing waves under ice. Using the more efficient code, we will invoke a more physical model to describe the interaction, including inhomogeneities of ice as well as dissipation of energy. In order to examine how physical solutions to the resulting equations are, we will modify existing methodology for stability analysis to also include dissipative effects. Since more extreme waves will occur in a nonlinear regime, we will also extend the stability analysis to capture these types of solutions. The impact of this work will not only contribute to understanding waves under ice, but the methodology developed will be applicable on a wide range of disciplines, whenever modelling of physical phenomena is done via partial differential equations.