A finite element approach for solving the shallow water equation on the sphere

This diploma thesis presents the development and implementation of a finite element method for solving the nonlinear shallow water equations on the sphere. Starting from a derivation of the shallow water equations from the incompressible Navier-Stokes equations, both the vector-invariant and conservative formulations are examined in terms of their conservation properties, numerical stability, and suitability for finite element discretisation. The method employs a compatible finite element framework with elements and spaces on spherical grids, specifically the Cubed- and Triangular Sphere, addressing geometric challenges such as coordinate singularities and metric consistency. Numerical experiments using standard benchmarks, including Galewsky flow and Rossby-Haurwitz waves, validate the accuracy and efficiency of the approach. The implementation is carried out in Julia, building upon existing numerical frameworks. The results support the method's potential for extension to more complex geophysical models involving additional physical processes.