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- Alina Chertock, Alexander Kurganov
- Numerische Mathematik
- 2008

The paper is concerned with development of 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… (More)

- G. I. Barenblatt, Michiel Bertsch, Alina Chertock, Valeriy 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)

Temporal, or “strict,” stability of approximation to PDEs is much more difficult to achieve than the “classical” Lax stability. In this paper, we present a class of finitedifference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability. The approximations are constructed so that… (More)

We compute multivalued solutions of oneand two-dimensional pressureless 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)

- Alina Chertock, Doron Levy
- 2001

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… (More)

We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS) system. Our modification is based on a fundamental physical property of the chemotactic flux function—its boundedness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient only when the latter is small,… (More)

This paper is devoted to a practical implementation of deterministic particle methods for solving transport equations with discontinuous 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)

Abstract. 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… (More)

- 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.