Boris Andreianov

ANALYSIS OF A FINITE VOLUME METHOD FOR A CROSS-DIFFUSION MODEL IN POPULATION DYNAMICS

The main goal of this work is to propose a convergent finite volume method for a reaction-diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time

Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks’ medium

L^1

Non-uniqueness of weak solutions for the fractal Burgers equation

compactness argument that mimics the compactness lemma due to S.N.~Kruzhkov. The proofs of these results are given in the Appendix.

Finite volume methods for degenerate chemotaxis model

For the hyperbolic conservation laws with discontinuous-flux function, there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley–Leverett equations, each notion of solution is uniquely determined by the choice of a “connection,” which is the unique stationary solution that takes the form of an under-compressive shock at the interface. To select the appropriate connection, following Kaasschieter (Comput Geosci 3(1):23–48, 1999), we use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in Cancès (Networks Het Media 5(3):635–647, 2010) that the “optimal” connection and the “barrier” connection may appear at the vanishing capillarity limit, we show that the intermediate connections can be relevant and the right notion of solution depends on the physical configuration. In particular, we stress the fact that the “optimal” entropy condition is not always the appropriate one (contrarily to the erroneous interpretation of Kaasschieter’s results which is sometimes encountered in the literature). We give a simple procedure that permits to determine the appropriate connection in terms of the flux profiles and capillary pressure profiles present in the model. This information is used to construct a finite volume numerical method for the Buckley–Leverett equation with interface coupling that retains information from the vanishing capillarity model. We support the theoretical result with numerical examples that illustrate the high efficiency of the algorithm.

Well-posedness of general boundary-value problems for scalar conservation laws

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the~

Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

L^\infty

Crowd dynamics and conservation laws with nonlocal constraints and capacity drop

-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.

A second-order model for vehicular traffics with local point constraints on the flow

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection-diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.

CONVERGENCE OF DISCRETE DUALITY FINITE VOLUME SCHEMES FOR THE CARDIAC BIDOMAIN MODEL

In this paper we investigate well-posedness for the problem

On 3D DDFV Discretization of Gradient and Divergence Operators:Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems

u_t+ \div \ph(u)=f