| In this work we present a new numerical method for shallow water equations on the surface of a rotating sphere. The method is based on the following idea. Place N points on the surface of a sphere to represent the fluid. Each point represents all the fluid that is closer to this point than to any other. This gives a network of polygonal cells called a Voronoi diagram. The principle of least action yields a set of difference equations which can be easily solved at each time step to yield new values for the height and velocity of the fluid particles. The points are then moved, using the new velocity, to their new locations and the Voronoi diagram is then reconstructed. In the event that two points get too close to each other, we consider the points to have collided. The collision is treated as an inelastic collision and the two points are merged into one, according to some conservation laws.; There are several advantages to this method. First, there is no dependence on coordinate systems and thus we can treat flows on a sphere without worrying about artificial singularities at the poles. Second, unlike other Lagrangian schemes, there is no restriction that points have to stay close to their initial neighbors. 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 if one fluid particle overtakes another one and collides with it. Thus, by merging two points into one, when they get too close, we keep the numerical scheme stable as well as handle shock waves.; The above scheme has been implemented on a CDC 6600 computer for the case of flow on a rotating sphere and for the one dimensional case. In one dimension we consider the dam-breaking problem, and get very good agreement with the exact solution. On the sphere we have treated an analogous problem with good results, although the exact solution on the sphere is not known. Other test cases have also been tried, with good results. |
