Technische Universität Darmstadt Fachbereich Mathematik AG Numerik und wissenschaftliches Rechnen
 Technische Universität Darmstadt Fachbereich Mathematik   Alf Gerisch

Numerical Methods for the Simulation of Taxis-Diffusion-Reaction Systems

by    Alf Gerisch

PhD thesis, Martin-Luther-Universität Halle-Wittenberg, Fachbereich Mathematik und Informatik, Institut für Numerische Mathematik, 23.08.2001.

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Abstract: We describe and evaluate a method of lines (MOL) technique for the simulation of taxis-diffusion-reaction (TDR) systems. These time-dependent PDE systems arise when modelling the spatio-temporal evolution of a population of organisms which migrate in direct response to e.g. concentration differences of a diffusible chemical in their surrounding (chemotaxis). Examples include pattern formation and different processes in cancer development. The effect of taxis is modelled by a nonlinear advection term in the TDR system (the taxis term). The MOL-ODE is obtained by replacing the spatial derivatives in the TDR system by finite volume approximations. These respect the conservation of mass property of the TDR system, and are constructed such that the MOL-ODE has a nonnegative analytic solution (positivity). The latter property is natural (because densities/concentrations are modelled) and highly desirable (because negative solution values might turn stable reaction terms into unstable ones). Diffusion and reaction terms can be replaced by standard approximations to ensure positivity, and we employ upwinding in combination with limiter functions in the discretization of the taxis term to ensure positivity of the MOL-ODE. The discretization near the boundary of the spatial domain is discussed. The appropriateness of the spatial discretization is demonstrated for a simple taxis problem (we provide the exact PDE solution). The MOL-ODE is stiff and of large dimension. We develop integration schemes which treat the discretization of taxis and diffusion/reaction differently (splitting). We employ operator (Strang-)splitting and/or the approximate matrix factorization technique. The splitting schemes are based on explicit Runge-Kutta (ERK) and linearly-implicit W-methods. Positivity and stability of the integration schemes are investigated. We identify an ERK method with favourable positivity properties. A corresponding W-method is constructed. Numerical experiments with a variety of splitting schemes applied to some semi-discretized TDR systems confirm the broad applicability of the splitting schemes and lead to a selection of efficient methods for the class of TDR systems. These methods are more efficient than (suitable) standard ODE solvers in the lower and moderate accuracy range. Altogether, the numerical technique developed is appropriate and efficient for the simulation of TDR systems.

MSC(2000):
65M20 Method of lines
92C15 Developmental biology, pattern formation

Keywords: Taxis-Diffusion-Reaction System, Mathematical Biology, Partial Differential Equation, Method of Lines, Finite Volume Method, Positivity, Operator Splitting, Rosenbrock Method, Approximate Matrix Factorization, Explicit Runge-Kutta Method

Notes:

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 Last Modification: 06.09.2010 17:21 Author: Alf Gerisch