A study of blow-ups in the Keller-Segel model of chemotaxis

We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then compared with analytical predictions regarding formation and interaction of singularities obtained via analysis of the stochastic differential equations associated with the Keller-Segel model

A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds

In this article we introduce a FCT stabilized Radial Basis Function (RBF)-Finite Difference (FD) method for the numerical solution of convection dominated problems. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the solution for an almost random placement of scattered data nodes. The method can be applicable both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable

Bound-preserving finite element approximations of the Keller-Segel equations

This paper aims to develop numerical approximations of the Keller--Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a nonnegative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear.Both algorithms can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and non-blowup phenomena. In the blowup setting, we identify a \textit{locking} phenomenon that relates the $L^\infty(\Omega)$-norm to the $L^1(\Omega)$-norm limiting the growth of the singularity when supported on a macroelement.Comment: 27 pages, 22 figure

Analysis of a fully discrete approximation for the classical Keller--Segel model: lower and a priori bounds

This paper is devoted to constructing approximate solutions for the classical Keller--Segel model governing \emph{chemotaxis}. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant. The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds -- positivity for the cell density and nonnegativity for the chemoattractant density --, are bounded in the $L^1(\Omega)$-norm, satisfy a discrete energy law, and have \emph{ a priori} energy estimates. The latter is achieved by means of a discrete Moser--Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.Comment: 24 pages, 6 figure