We describe 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).

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 and linearly-implicit W-methods. Results on the positivity and the stability of integration schemes are discussed.

Numerical experiments with a variety of splitting schemes applied to some semi-discretized TDR systems confirm the broad applicability of the splitting schemes. 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.