Particle Methods for Dispersive Equations

We introduce a new dispersion-velocity particle method for approximating solutions of linear and nonlinear dispersive equations. This is the first time in which particle methods are being used for solving such equations. Our method is based on an extension of the diffusion-velocity method of Degond and Mustieles (SIAM J. Sci. Stat. Comput.11(2), 293 (1990)) to the dispersive framework. The main analytical result we provide is the short time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. We numerically test our new method for a variety of linear and nonlinear problems. In particular we are interested in nonlinear equations which generate structures that have nonsmooth fronts. Our simulations show that this particle method is capable of capturing the nonlinear regime of a compacton–compacton type interaction.