Advanced numerical treatment of chemotaxis driven PDEs in mathematical biology

Abstract

From the first formulation of chemotaxis-driven partial differential equations (PDEs) by Keller and Segel in the 1970's up to the present, much effort has been expended in modelling complex chemotaxis re- lated processes. The shear complexity of such resulting PDEs crucially limits the postulation of analytical results. In this context the sup- port by numerical tools are of utmost interest and, thus, render the implementation of a numerically well elaborated solver an undoubt- edly important task. In this work I present different iteration strategies (linear/nonlinear, decoupled/monolithic) for chemotaxis-driven PDEs. The discretiza- tion follows the method of lines, where I employ finite elements to resolve the spatial discretization. I extensively study the numerical efficiency of the iteration strategies by applying them on particular chemotaxis models. Moreover, I demonstrate the need of numerical stabilization of chemotaxis-driven PDEs and apply a exible scalar algebraic ux correction. This methodology preserves the positivity of the fully discretized scheme under mild conditions and renders the numerical solution non-oscillatory at a low level of additional compu- tational costs. This work provides a first detailed study of accurate, efficient and exible finite element schemes for chemotaxis-driven PDEs and the implemented numerical framework provides a valuable basis for fu- ture applications of the solvers to more complex models.