Methods for Solving Nonlinear Differential Equations Describing Water Waves

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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.