Geometric numerical integration –in geophysical and environmental fluid dynamics

Geometric numerical integration concerns the preservations the geometric and conservative structure of the underlying PDEs. This geometric structure is responsible for the (integral) conservation laws of the PDEs. Such conservation laws can include: mass, energy, phase-space volume, and potential vorticity. For PDEs with a variational or Hamiltonian structure, conservation laws are a consequence of Noether's theorem. The great and appealing advantage of geometric numerical integration is to preserve these conservation laws as much as possible in order to ensure faithful integration, numerical stability and robustness. After initial explorations of the literature and some example systems, we encourage you (the PhD candidate) to focus on a particular system of interest. Exciting systems of interest, with completely different application areas, could be balanced models for rapidly rotating Rayleigh-Bénard convection (relevant to geophysical fluid dynamics) or surface water waves combined with shallow-water vorticity, the latter allowing the combined effects of dispersive waves and breaking waves or hydraulic bores (relevant to maritime and coastal engineering).

Figure: Simulation of a rogue wave, seen to peak at time 109.40s, using a geometric numerical integration of water waves driven by a piston wave-maker (Courtesy Floriane Gidel, PhD thesis, University of Leeds, 2018). The domain with its wavy free surface a well as the velocity potential are shown.