We extend the dispersion-velocity particle method that we recently introduced to advection models in which the velocity does not depend linearly on the solution or its derivatives. An example is the Korteweg de Vries (KdV) equation for which we derive a particle method and demonstrate numerically how it captures soliton–soliton interactions.
The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety… (More)
Recently, a wavelet-based method was introduced for the systematic derivation of subgrid scale models in the numerical solution of partial differential equations. Starting from a discretization of the multiscale differential operator, the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The coarse (homogenized)… (More)
The dispersal and mixing of scalar quantities such as concentrations or thermal energy are often modeled by advection-diffusion equations. Such problems arise in a wide variety of engineering, ecological and geophysical applications. In these situations a quantity such as chemical or pollutant concentration or temperature variation diffuses while being… (More)
We first present a new sticky particle method for the system of pressureless gas dynamics. The method is based on the idea of sticky particles, which seems to work perfectly well for the models with point mass concentrations and strong singularity formations. In this method, the solution is sought in the form of a linear combination of δ-functions, whose… (More)
We present a new hybrid numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. The idea behind the new finite-volume-particle (FVP) method is to use different schemes for the flow and the… (More)