Particle Methods for Some Nonlinear PDEs

Alina Chertock, Department of Mathematics, North Carolina State University

In recent years, particle methods have become one of the most useful and widespread tools for approximating solutions of partial differential equations. They have been successfully used to treat a broad class of problems arising in astrophysics, plasma physics, solid state physics, medical physics, and fluid dynamics. In these methods, a solution of a given equation is represented by a collection of particles, located at points $x_{i}$ and carrying masses $w_{i}$. At later times, the locations of the particles and/or their weights are allowed to change. The solution is then found by following the time evolution of the locations and of the weights of the particles according to a transport equation. Due to the Lagrangian nature of the method, small scales that might develop in a solution can be easily described with a relatively small number of particles. This property is what made particle methods so attractive in practice.

In this talk, I will present a new particle method for approximating solutions of linear and nonlinear dispersive equations. This is the first time that particle methods are being applied to this type of equations. Our method is based on the diffusion-velocity method, which was introduced by Degond and Mustieles [SIAM J. Sci. Stat. Comp., 11, (1990), pp.293--310] for approximating solutions of parabolic equations. The main analytical result we provide is the short time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. The new particle method has been implemented for variety of linear and nonlinear problems. Among other examples are so-called $K(m,n)$ type equations, which generate compactly supported solutions (compactons) with non-smooth fronts, and the KdV equation. It should be pointed out that this was the first attempt ever to use particles for the direct simulation of solitary wave interactions. Numerical experiments show that our particle method is capable of capturing the nonlinear regime of a compacton-compacton and a soliton-soliton interaction.

Areas of applications of the new method are not restricted to dispersive equations. I will also discus how to apply the method to degenerate parabolic equations with saturated fluxes.

This is a joint work with Doron Levy, Stanford University, and Alexander Kurganov, Tulane University.