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This paper is devoted to a study of the asymptotic behavior of solutions of a chemotaxis model with logistic terms in multiple spatial dimensions. Of particular interest is the practically relevant case of small diffusivity, where (as in the one-dimensional case) the cell densities form plateau-like solutions for large time.
The major difference from the one-dimensional case is the motion of these plateau-like solutions with respect to the plateau boundaries separating zero density regions from maximum density regions. This interface motion appears on a non-logarithmic time scale and can be interpreted as a surface diffusion law. The biological interpretation of the surface diffusion is that a packed region of cells can change its shape mainly if cells diffuse along its boundary.
The theoretical results on the asymptotic behavior are supplemented by several numerical simulations on two- and three-dimensional domains.
In this paper, we obtain new models for gravity driven shallow water laminar flows in several space dimensions over a general topography. These models are derived from the incompressible Navier-Stokes equations with no-slip condition at the bottom and include capillary effects. No particular assumption is made on the size of the viscosity and on the variations of the slope. The equations are written for an arbitrary parametrization of the bottom, and an explicit formulation is given in the orthogonal courvilinear coordinates setting and for a particular parametrization so-called “steepest descent” curvilinear coordinates.
We study the numerical approximation of scalar conservation laws in dimension 1 via general reconstruction schemes within the finite volume framework. We exhibit a new stability condition, derived from an analysis of the spatial convolutions of entropy solutions with characteristic functions of intervals. We then propose a criterion that ensures the existence of some numerical entropy fluxes. The consequence is the convergence of the approximate solution to the unique entropy solution of the considered equation.
To study the numerical solutions of quasilinear elliptic equations on unbounded domains in two or three dimensional cases, we introduce a circular or spherical artificial boundary. Based on the Kirchhoff transformation and the Fourier series expansion, the exact artificial boundary condition and a series of its approximations of the given quasilinear elliptic problem are presented. Then the original problem is equivalently or approximately reduced to a bounded computational domain. The well-posedness of the reduced problems are proved and the convergence results of our numerical solutions on bounded computational domain are given
Variational descriptions for various multiphase level set formulations involving curvature flow are discussed. A representation of $n$ phases using $n−1$ level set functions is introduced having the advantage that constraints preventing overlaps or vacuum are not needed. The representation is then used in conjunction with our variational formulation to deduce a novel level set based algorithm for multiphase flow. In addition, a similar variational formulation is applied to area preserving curvature flow. In this flow, the area (or volume in 3D) enclosed by each level set is preserved. Each algorithm has been implemented numerically and the results of such computations are shown.
This work is devoted to the study of the system that arises by discretization of the periodic nonlinear Schrödinger equation in dimension one. We study the existence of the discrete ground states for this system and their stability property when the potential parameter ¾ is small enough: i.e., if the initial data are close to the ground state, the solution of the system will remain near to the orbit of the discrete ground state forever. This stability property is an appropriate tool for proving the convergence of the numerical method.
In this paper, we discuss the multicommodity flow for vehicular traffic on road networks. To model the traffic, we use the “Aw-Rascle” multiclass macroscopic model. We describe a solution to the Riemann problem at junctions with a criterion of maximization of the total flux, taking into account the destination path of the vehicles. At such a junction, the actual distribution depends on the demands and the supplies on the incoming and outgoing roads, respectively. Furthermore, this new distribution scheme captures efficiently key merging characteristics of the traffic and in contrast to M. Herty, S. Moutari and M. Rascle, Networks and Heterogeneous Media, 1, 275-294, 2006, leads to an easy computational model to solve approximately the homogenization problem described in M. Herty, S. Moutari and M. Rascle, Networks and Heterogeneous Media, 1, 275-294, 2006, (M. Herty and M. Rascle, SIAM J. Math. Anal., 38(2), 595-616, 2006). Furthermore, we deduce the equivalent distribution scheme for the LWR multiclass model in M. Garavello and B. Piccoli, Commun. Math. Sci., 3, 261-283, 2005, and we compare the results with those obtained with the “Aw-Rascle” multiclass model for the same initial conditions.
In this paper, we state a convergence result for an $L1$-based finite element approximation technique in one dimension. The proof of this result is constructive and provides the basis for an algorithm for computing $L1$-based almost minimizers with optimal complexity. Several numerical results are presented to illustrate the performance of the method.