As a first step towards construction of a DG based dynamical core for high resolution atmospheric modelling, a semi-implicit and semi-Lagrangian discontinuous Galerkin method for the SWE on the sphere and for nonhydrostatic vertical slice equations is proposed and analysed. The method is equipped with a simple p-adaptivity criterion, that allows to adjust dynamically the number of degrees of fr...

We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we show that the number of nodal domains of the n-th eigenfunction is bounded below by n− l, where l is the number of links that distinguish the graph from a tree. Our results apply to operators on bo...