We present a particle method for studying a quasilinear partial differential equation (PDE) in a class proposed for the regularization of the Hopf (inviscid Burger) equation via nonlinear dispersion-like terms. These are obtained in an advection equation by coupling the advecting field to the advected one through a Helmholtz operator. Solutions of this PDE are "regularized" in the sense that the additional terms generated by the coupling prevent solution multivaluedness from occurring. We propose a particle algorithm to solve the quasilinear PDE. "Particles" in this algorithm travel along characteristic curves of the equation, and their positions and momenta determine the solution of the PDE. The algorithm follows the particle trajectories by integrating a pair of integro-differential equations that govern the evolution of particle positions and momenta. We introduce a fast summation algorithm that reduces the computational cost from O(N(2)) to O(N), where N is the number of particles, and illustrate the relation between dynamics of the momentum-like characteristic variable and the behavior of the solution of the PDE.
Communications on Pure and Applied Analysis
Camassa, R.; Chiu, P. H.; Lee, Long; and Sheu, T.W.H. (2011). "A Particle Method and Numerical Study of a Quasilinear Partial Differential Equation." Communications on Pure and Applied Analysis 10.5, 1503-1515.