FEEC discretizations · Structure-preserving discretization of hierarchically-structured rotating covariant shallow-water equations using finite element exterior calculus
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2016-04-01 → 2018-03-31
- EU contribution
- €183,455
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Structure-preserving discretization of hierarchically-structured rotating covariant shallow-water equations using finite element exterior calculus
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.
Data: CORDIS, © European Union
Project objective
Accurate and reliable simulations of weather, ocean and climate require computational models that result from structure-preserving – e.g. mass or energy conserving – discretizations of the equations of geophysical fluid dynamics (GFD). This research project aims to derive, implement and evaluate various structure-preserving discretizations (of different order of accuracy) of the nonlinear shallow-water equations, which are suitable for weather/ocean/climate applications. The derivations will rely on a novel form of covariant equations of GFD that I have formulated using Differential Geometry, in which the equations are split into metric-free (topological) and metric-dependent parts. Based on the systematic discretization I have introduced for the split linear shallow-water equations, this project intends to extend this approach also to the split nonlinear case and to derive structure-preserving discretizations that preserve in the discrete case, too, the splitting into topological and metric terms. As the topological terms require less mathematical structure, we expect an advantage in terms of easiness of discretization and efficiency of implementation.To derive corresponding discrete equations, we apply finite element exterior calculus (FEEC) as recently Cotter and Thuburn, whose resulting discretizations of conventional covariant nonlinear shallow-water equations fulfil many desirable properties for geophysical applications. Moreover, compared to the split form I proposed, their discrete equations show a similar, however not identical, structure. We study the differences and use their derivations as guideline for ours. To implement and test the various models, we use the software libraries Firedrake and FEniCS. Besides a general “discretization recipe” to derive structure-preserving models, this project will provide open-source software which will be of practical use for the geophysical model community.
Original text from CORDIS.
Participants
- IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE · LondonCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union