Chiara Sorgentone: Energy, enstrophy and symmetry preserving schemes for the numerical integration of non-linear advective problems

Arakawa's Jacobian is one of the most renowned scheme for the numerical integration of the vorticity equation for two-dimensional incompressible flow. Its peculiarity lies in the mimetic properties such as skew-symmetry, conservation of mean kinetic energy and mean square vorticity which can avoid non-linear numerical instabilities. I will study different ways to generalize Arakawa's idea: the first idea is to construct a general procedure for non-linear mimetic finite-difference operators. The method is very simple and general: it can be applied for any order scheme, for any number of grid points and for any operator constraints. I chose the specific set of second order mimetic schemes with compact stencil in order to compare my solution with the Arakawa's one (which result to be a specific case of the general solution). I also developed high-order mimetic Arakawa-like scheme using Summation-by-Parts (SBP) operators and I will prove mimetic properties in this new abstract and general space. In addition to analytical test examples, a geophysical application will also be presented: the study of a two-layer quasi-geostrophic model.