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Structure-preserving discretization of hierarchically-structured rotating covariant shallow-water equations using finite element exterior calculus

Periodic Reporting for period 1 - FEEC discretizations (Structure-preserving discretization of hierarchically-structured rotating covariant shallow-water equations using finite element exterior calculus)

Reporting period: 2016-04-01 to 2018-03-31

Accurate and reliable simulations of weather and climate require computational models that result from structure-preserving discretizations of the equations of geophysical fluid dynamics (GFD). The construction of such schemes though is not straight forward and ad hoc methods might fail to preserve important conservation properties. The project's objectives were to develop general methods in deriving structure-preserving discretizations for a large variety of equations, with a particular focus on the derivation, implementation, and evaluation of various structure-preserving discretizations (of different order of accuracy) of the rotating shallow water (RSW) equations suitable for atmosphere, ocean, and climate applications.

In this project, we developed two alternative discretization approaches: (i) the split finite element (FE) framework that is based on the splitting of the equations into topological and metric parts (Bauer 2016), and (ii) a variational discretization framework for compressible fluids that is based on variational principles. As an extension of the PI's original framework developed for the linear shallow-water equations, the current split FE framework for the RSW equations provides a systematic method to derive structure-preserving discretizations that preserve the split structure. The variational discretization framework applies discrete variational principles to derive discrete equations of motion for a given discrete Lagrangian by Hamilton's principle of least action. In this vein, we extended an existing theory for incompressible fluids to compressible fluids and derived and evaluated variational integrators for the RSW equations.
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) and one about the compressible RSW equations (https://arxiv.org/pdf/1711.10617.pdf) 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).

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 https://arxiv.org/pdf/1801.00691.pdf 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 c
We developed two alternative approaches to derive structure-preserving discretizations for various PDEs, in particular for the equations of GFD. We are convinced that this contribution will have a significant impact on the development of upcoming conservative, efficient and accurate computational models, in particular in the current discussion about climate change, where accurate long term predictions are essential to develop mitigation and adaptation strategies.

The variational framework provides the, to the best of our knowledge, first fully consistent variational discretization in space and time of compressible fluids leading to more accurate structure-preserving schemes that carry useful mathematical structure to enhance the code performance and allow for automatic code generation; a feature which becomes more and more important for operational centers, such as the Metoffice UK.

The split FE framework provides a novel FE method based on FEEC. These schemes preserve the clear separation of structure-preserving prognostic equations from metric-dependent closure equations, the latter allow for approximations that do not impact on the structure preservation while preserving local flow features.

Finally, the idea of the split equations has a significant impact on the PI's carrier. Not only was this idea the foundation of this Marie Skłodowska-Curie project, it is also the basis for several collaborations.
Dispersion relations for the split FE schemes
PI Werner Bauer
Commutative diagram for the split FE schemes