Discontinuous Galerkin methods for the chemotaxis and haptotaxis models
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
Journal of Scientific Computing
A Finite Volume Method for Solving Parabolic Equations on Logically Cartesian Curved Surface Meshes
SIAM Journal on Scientific Computing
Efficient, accurate and flexible finite element solvers for chemotaxis problems
Computers & Mathematics with Applications
Computer modelling of haematopoietic stem cells migration
Computers & Mathematics with Applications
Maxwellian Decay for Well-balanced Approximations of a Super-characteristic Chemotaxis Model
SIAM Journal on Scientific Computing
A positivity-preserving finite element method for chemotaxis problems in 3D
Journal of Computational and Applied Mathematics
Upwind-Difference Potentials Method for Patlak-Keller-Segel Chemotaxis Model
Journal of Scientific Computing
Nonnegativity of exact and numerical solutions of some chemotactic models
Computers & Mathematics with Applications
Hi-index | 0.01 |
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 concentration. The first step in the derivation of the new method is made by adding an equation for the chemoattractant concentration gradient to the original system. We then show that the convective part of the resulting system is typically of a mixed hyperbolic-elliptic type and therefore straightforward numerical methods for the studied system may be unstable. The proposed method is based on the application of the second-order central-upwind scheme, originally developed for hyperbolic systems of conservation laws in Kurganov et al. (SIAM J Sci Comput 21:707–740, 2001), to the extended system of PDEs. We show that the proposed second-order scheme is positivity preserving, which is a very important stability property of the method. The scheme is applied to a number of two-dimensional problems including the most commonly used Keller–Segel chemotaxis model and its modern extensions as well as to a haptotaxis system modeling tumor invasion into surrounding healthy tissue. Our numerical results demonstrate high accuracy, stability, and robustness of the proposed scheme.