Structure Preserving Discretization of the Rotating Shallow Water Equations
Colorado State University
The shallow water equations provide a useful analog of fully compressible Euler equations since they have similar conservation laws, many of the same types of waves, and a similar (quasi-) balanced state. With regard to conservation properties, there have been two major thrusts of research: Hamiltonian methods (work done by Salmon and Dubois) and Discrete Exterior Calculus (DEC; Thuburn, Cotter, Ringler, etc.). In particular, recent work by Thuburn and Cotter introduced a generalized framework for energy-conservative C-grid discretizations of the rotating shallow water equation using ideas from Discrete Exterior Calculus. The current research elucidates the connections between the Hamiltonian and DEC approaches and looks at potential enstrophy conservation in addition to energy conservation. Finally, a generalized framework for mimetic total energy and potential enstrophy-conserving discretizations of the rotating shallow water equation in vorticity-divergence form (also using the DEC approach) is developed.