A Lagrangian method for the shallow water equations based on a Voronoi mesh — Flows on a rotating sphere

Abstract

We have presented in this work and in a previous work [1, 2] a promising new Lagrangian method for the shallow water equations. The advantages of this method may be summarized as follows. First, there is no dependence on coordinate systems, and thus we can treat flows on a sphere without worrying about pole singularities. Second, unlike other Lagrangian schemes, there is no restriction that points have to retain their initial neighbors. On the contrary, at each time step the particles find their natural neighbors, and the derivatives are computed using these neighbors. Thus the method allows for large deformations. Third, a novel feature of the method is that it handles shocks in a very natural way. We consider a shock as one fluid particle overtaking another and colliding with it. This procedure not only makes it possible to handle shocks (which are unimportant in atmospheric flow calculations since the rotation of the earth causes waves to be dispersive) but it also guarantees the stability of the scheme by enforcing the Courant-Friedrichs-Lewy conditions in the neighborhood of each fluid marker. Fourth, since the markers can be placed anywhere, the fluid markers may be placed, initially, at points where satellite or other measured data are available. The main problem with Free Lagrangian methods however, is that the discrete operators seem to be of low order accuracy. Further research is needed to derive high order accurate schemes to compete with existing Eulerian schemes.