Computing highly accurate solutions of a tumour angiogenesis model

In this paper we describe and implement a numerical method which provides highly accurate solutions of a generic two-dimensional model for the formation of capillary networks as a partial process in tumour angiogenesis. The model includes effects due to diffusion, chemotaxis, haptotaxis and cell proliferation. The governing partial differential equation is a diffusion-advection-reaction equation of parabolic type. In order to achieve high accuracy in space, we use a semi-discretisation based on the spectral element method. The resulting system of stiff ordinary differential equations is advanced forward in time using one-step explicit higher order time integrators based on Taylor series expansions. The high accuracy in space is monitored by a residual based a posteriori error indicator while the high accuracy in time is guaranteed by the local and global truncation errors of the higher order Taylor series method.