A formulation with Exterior Calculus (EC) gives us a deeper insight into the geometrical structure of the Euler equations. Our aim is to compare this formulation with the Energy-Vorticity-Theory (ETV) developed by Peter Névir. By using Discrete Exterior Calculus (DEC), we are going to investigate several methods on how to define the geometrical operators on the ICON-grid.
We computed a general and comprehensive derivation of the Euler equations in Exterior Calculus. In the following as an example, we present the shallow-water equations:
More information on shallow water equations in Exterior Calculus can be found in Juan Simarro's talk: Exterior Calculus in simulation of fluids
We are gaining a better understanding for the geometrical structure of fluid dynamics by using an invariant formulation as in EC.
Calculations, derivations and proofs can be shortend by using the modern language of differential geometry. We have verified the connection between several notions of the conventional Vector Calculus and Exterior Calculus.
Using Whitneyforms as interpolation functions we are able to write the shallow-water equations in a weak formulation and apply standard numerical methods, for example Galerkin's method.
We are aiming to develop a systematical way of discretizing Euler's equations by using Discrete Exterior Calculus (DEC) by Anil Hirani. Therefore, we are developing an algorithm on how to discretize Euler's equations on the ICON-grid under the constraint of energy and enstrophy conservation. In addition, the properties of operators on continuous manifolds should also be fulfilled in the discretized case. Finally, we compare this approach with the Energy-Vorticity-Theory (ETV) developed by Peter Nevir [Nevir, P., Meteorologische Zeitschrift, Volume 13, 2005].
Bossavit et. al. [Bossavit, A., SIAM J. NUMER. ANAL., 2005] developed prolongation and restriction operators for multigrid-methods used for interpolating information between different grid levels. These operators conserve the properties of differential operators on different grid levels. We will apply these tools on Euler's equations, formulated in EC, for grid refinement strategies on the ICON-grid.
(wba, 27.10.2008)