A semi-implicit, semi-Lagrangian, p-adaptive Discontinuous Galerkinmethod for the shallow water equations

As a first step in the context of a broader work concerning the construction and analysis of a new generation DG based dynamical core for atmospheric modelling, a semi-implicit and semi-Lagrangian Discontinuous Galerkin method for the shallow water equations is proposed and analyzed in the one dimensional case.
The shallow water equations actually contain all of the horizontal operators required in an atmospheric model and thus represent a good first test for newly proposed schemes for atmospheric
simulations.
The method is equipped with a simple p-adaptivity
criterion, that allows to adjust effectively the number of local degrees of freedom employed to the local structure of the solution.
Numerical results in the framework of one dimensional test cases prove that the method captures accurately and effectively the main features of linear gravity and inertial gravity
waves, as well as reproducing correct solutions in nonlinear open channel flow tests.
The effectiveness of the method is also demonstrated by numerical results obtained
at high Courant numbers and with automatic choice of the local approximation degree.
Perspectives for extensions to multidimensional applications are discussed. Finally some results are shown about conservative passive tracers
advection, this issue being particularly important for climate modelling applications
(where a lot of chemical species are advected with no influence on the dynamics of the flow).