A novel numerical solver for nonlinear boundary value problems, with applications to the forced Gardner equation

Nonlinear differential equations appear within a wide range of fields, however, solving these problems is particularly difficult. This work charts the development of a suite of new, numerical tools for solving nonlinear differential equations. These tools significantly outperform currently existing methods, in both the amount of time required to solve the problems, and the range of problems that can be approached. After analysing the properties of these tools, they are applied to the forced Gardner equation. This problem arises in geophysical fluid dynamics, and has particular implications to climate change and extreme weather events.