Variational derivation of energy-conserving finite-difference schemes for geophysical fluid equations
Seminar Room 1, Newton Institute
At the continuous level, the so-called vector-invariant form of the equations of fluid motion can be obtained from Hamilton's principle of least action, using the Euler-Poincaré formalism. In this form of the equations, the mass flux and Bernoulli function appear as functional derivatives of the Hamiltonian. The integral conservation of energy and the Lagrangian conservation of potential vorticity then follow straightforwardly.
This setting can be imitated at the discrete level to yield energy-conserving schemes. Key ingredients include an energy-conserving vector product, a discrete rule of integration by parts, a discrete approximation of total energy, and the definition of the discrete mass flux and Bernoulli function from partial derivatives of the discrete energy. For the rotating shallow-water equations, it turns out that schemes by Sadourny (1975) on Cartesian meshes and by Bonaventura & Ringler (2005) on Delaunay-Voronoi meshes fit in the above framework.
Furthermore new schemes can be obtained. For instance it is possible to modify the discrete mass flux to match a modification of the discrete energy, as Renner (1981) and Skamarock et al. (2012) did to suppress a numerical instability. Finally, a so-called discrete Hodge star operator is obtained on a non-orthogonal pair of primal and dual meshes. This operator is part of a potential-vorticity conserving scheme of the shallow-water equations on meshes for which an orthogonal dual does not exist, like the equiangular cubed sphere. Potential application to sets of three-dimensional equations will be discussed.