In collaboration with Dr F. Gay-Balmaz (CNRS/LMD-ENS, Paris, France), the PI developed structure-preserving discretizations of the RSW equations based on variational principles. The results are an extension of their former derivations of variational integrators for soundproof approximations of the Euler equations and provide a further extension of the original variational discretization framework of Pavlow et al. (2009) for incompressible fluids. The key point is the identification of the appropriate discrete Lie group that approximate the corresponding configuration space of compressible fluids. This framework easily extends to other fields of interest such as magnetohydrodynamics and electrodynamics. This result is a milestone in deriving structure-preserving discretizations, as it provides a complete framework to derive consistent schemes for compressible fluids fully from variational principles. The work resulted in two preprints on arxiv.org one about variational integrators for soundproof models (
https://arxiv.org/pdf/1701.06448.pdf(odnośnik otworzy się w nowym oknie)) and one about the compressible RSW equations (
https://arxiv.org/pdf/1711.10617.pdf(odnośnik otworzy się w nowym oknie)) which have been submitted for publication. A further article is in preparation.
In collaboration with Prof J. Behrens (Universität Hamburg, Germany) and Dr C. Cotter (Imperial College London, UK), the PI developed the split FE framework on the basis of the split equations of GFD. The framework allows for a mathematically clean formulation and discrete preservation of the structure of the underlying partial differential equation (PDE) by decoupling the discretization into topological and metric-dependent equations. While the topological equations preserve structure (conservation of mass, momentum, energy), the metric-dependent equations allow for a flexible choice of FE spaces. We introduced this framework for the linear 1D split wave equations. This novel FE method has been published in Applied Mathematics and Computations (AMC) (
https://doi.org/10.1016/j.amc.2017.12.035(odnośnik otworzy się w nowym oknie)).
Moreover, we worked on extensions of the split FE framework to include the RSW equations using the Hamiltonian form and Finite Element Exterior Calculus (FEEC). For various schemes realized with the software library FIREDRAKE, we studied if the flow is locally conserved, in contrast to classical FE approaches where locality in the flow usually cannot be guaranteed. These results are content of two papers that are currently in preparation.
In collaboration with Dr C. Cotter and based on the idea of splitting, we could even solve erroneous treatment of boundary conditions in an existing FE shallow water scheme. The key idea is to introduce an additional prognostic variable for the potential vorticity rather than only prognosing it. This allows for a consistent energy and potential enstrophy conserving FE scheme in case of domains with boundaries. The paper is available as preprint on arxiv.org (
https://arxiv.org/pdf/1801.00691.pdf(odnośnik otworzy się w nowym oknie)) and is currently under review at JCP.
With C. Eldred (INRIA-Grenoble, France), the PI generalized the split equations of GFD towards a split Hamiltonian form with topological Poisson brackets and metric-dependent Hamiltonian. These findings allowed us to understand TRISK, a frequently used C-grid discretization of the RSW equations on polygonal meshes with many desirable conservation properties (e.g. steady geostrophic modes, conservation of energy), fully in terms of Discrete Exterior Calculus. In particular, we could associate the scheme's conservation properties with a Leibnitz rule of the discrete wedge product that is used to represent the nonlinear terms. Together with Prof J. Thuburn (University of Exeter, UK), we are currently addressing the accuracy issue of TRISK showing inconsistencies in the nonlinear terms.
Finally, with Dr A. Bihlo and colleagues from the Memorial University of Newfoundland, Canada, we developed variational integrators for the RSW equations on the sphere and we are going to combine them with a coastal inundation model. A paper about the numerical performance of the variational RSW scheme in spherical geometry is currently in preparation.
