  We present an accurate and efficient discretization approach for the adaptive
discretization of typical model equations employed in numerical weather
prediction. A semi-Lagrangian approach is combined with the TR-BDF2
semi-implicit time discretization method and with a spatial discretization
based on adaptive discontinuous finite elements. The resulting method has full
second order accuracy in time and can employ polynomial bases of arbitrarily
high degree in space, is unconditionally stable and can effectively adapt the
number of degrees of freedom employed in each element, in order to balance
accuracy and computational cost. The p-adaptivity approach employed does not
require remeshing, therefore it is especially suitable for applications, such
as numerical weather prediction, in which a large number of physical quantities
are associated with a given mesh. Furthermore, although the proposed method can
be implemented on arbitrary unstructured and nonconforming meshes, even its
application on simple Cartesian meshes in spherical coordinates can cure
effectively the pole problem by reducing the polynomial degree used in the
polar elements. Numerical simulations of classical benchmarks for the shallow
water and for the fully compressible Euler equations validate the method and
demonstrate its capability to achieve accurate results also at large Courant
numbers, with time steps up to 100 times larger than those of typical explicit
discretizations of the same problems, while reducing the computational cost
thanks to the adaptivity algorithm.
