Lagrangian method for the shallow water equations based on a Voronoi mesh--one dimensional results

A Lagrangian method for the Shallow Water equations in one dimension is presented. The method is based on the following idea. Place N points on a line to represent the fluid. Each point represent all the fluid that is closer to this point than to any other. This partitions the line into intervals. The principle of Least Action yields a set of difference equations which can be easily solved at each time step to give new values for the height and velocity of the fluid particles in each interval. The points are then moved to their new locations and the intervals reconstructed. In the event that two points get too close to each other, consider the points to have collided. The collision is treated as an inelastic collision and the two points are merged into one, according to certain conservation laws. It is then possible to handle flows with shocks by modelling a shock as a collision between two particles. That is, a shock is thought of as one fluid particle overtaking another one and colliding with it. The results of this method on the Dam-Breaking problem are presented. The solution is compared to the exact solution and to a solution by the Random Choice Method