Energy conserving compatible space--time FEM for deterministic and stochastic rotating shallow water

Abstract

We present a fully discrete energy conserving discretisation of the rotating shallow water equations with arbitrary order in space and time. A space time finite element discretisation is used, built from compatible finite element methods in space and continuous Petrov-Galerkin schemes in time. Projected auxiliary variables for Hamiltonian derivatives are necessary for exact energy conservation. We provide an iterative solver approach based on monolithic patch smoothers, which allows us to efficiently converge a Newton method sufficiently that energy errors due to the solver are very small. We extend this construction to the energy conserving Stochastic Forcing by Lie Transport formulation. Then we demonstrate the convergence, conservation and robustness in numerical tests.