Robust numerical methods for taxis–diffusion–reaction systems: Applications to biomedical problems

In this paper we consider the numerical solution of taxis–diffusion–reaction models. Such nonlinear partial differential equation models appear often in mathematical biology and we consider examples from tumour growth and invasion, and the aggregation of amoebae. These examples are characterised by a taxis term which is large in magnitude compared with the diffusion term and this can cause numerical problems. The numerical technique presented here follows the method of lines. Special attention is paid to the discretization of the taxis term in space to avoid oscillations and negative solution values. We employ splitting techniques in the time discretization to deal with the complex structure of the model and to reduce the amount of computational linear algebra. These techniques are based on explicit Runge–Kutta and linearly implicit Runge–Kutta–Rosenbrock methods. A series of numerical experiments demonstrates the good performance of the algorithm and gives rise to some implications for future modelling.