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In this talk, I will present a new finite-volume method for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. A… (More)

- Alina Chertock, Alexander Kurganov
- Numerische Mathematik
- 2008

- Alina Chertock, Doron Levy
- J. Sci. Comput.
- 2002

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.

- Alina Chertock, Alexander Kurganov, Guergana Petrova
- J. Sci. Comput.
- 2006

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)

We compute multivalued solutions of one-and two-dimensional pressure-less gas dynamics equations by deterministic particle methods. Point values of the computed solutions are to be recovered from their singular particle approximations using some smoothing procedure. We study several recovery strategies and demonstrate ability of the particle methods to… (More)

- G I Barenblatt, M Bertsch, A E Chertock, V M Prostokishin
- Proceedings of the National Academy of Sciences…
- 2000

The equation partial differential(t)u = u partial differential(xx)(2)u -(c-1)( partial differential(x)u)(2) is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a… (More)

The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also… (More)

- Alina Chertock, Jian-Guo Liu, Terrance Pendleton
- SIAM J. Numerical Analysis
- 2012

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)

This paper is devoted to a practical implementation of deterministic particle methods for solving transport equations with dis-continuous coefficients and/or initial data, and related problems. In such methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights… (More)

- Alina Chertock, Alexander Kurganov, Yurii Rykov
- SIAM J. Numerical Analysis
- 2007

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)