Covariant equations of geophysical fluid dynamics and their structure-preserving discretization
Objectives and Key Questions
The accuracy and complexity of computational models used to simulate geophysical scenarios crucially depends on the analytical equations that describe the physical phenomena and on how these equations are discretized. A large variety of computational models exists in literature. This descends from the fact that different analytical equations, often written in terms of vector-calculus, are used to describe the different physical phenomena and that form, dimension and complexity of these equations may call for different discretization procedures.
Within this project we use the language of differential geometry and exterior calculus to formulate generalized covariant equations of geophysical fluid dynamics. In addition, we introduce a new structure for these equations. For instance, we split the rotating Euler's equations into a metric-free and a metric-dependent part, which hierarchically orders the equations with respect to the mathematical spaces required. In comparison to the form of conventional vector-invariant equations, this structure better reflects the geometrical properties of the fluid flow. Because of their dimensionless and coordinate-free formulation, these covariant equations are valid for a wide range of geophysical applications.
We are developing systematic discretization methods by using the tools of Discrete Exterior Calculus (DEC) and of Finite Element Exterior Calculus (FEEC), in which the form of the analytical equations, in particular their hierarchical structure, suggests how to discretize them in a structure-preserving manner. The resulting computational models guarantee accurate simulations, because structure-preserving discretizations inherit important properties of the analytical equations, such as mass and energy conservation and preservation of certain flow features.
Responsible group members
- Werner Bauer