Splitting Methods for the Simulation of Tumor Angiogenesis Models

Angiogenesis – the process by which new blood vessels grow into a tissue from surrounding parent vessels – is an important process in many areas of medicine. Here we consider the numerical simulation of a PDE model of tumor-induced angiogenesis. It contains convection (migration), diffusion and reaction terms. Despite the restriction to one specific model, the observations should also be relevant for the solution of similar problems. Our general approach is the method of lines and we use a positivity preserving spatial discretization resulting in a large and in general stiff ODE system. For the solution of this system we consider splitting methods (approximate matrix factorization, Strang-type and source splitting) and Krylov-W-methods. The aim is to reduce the complexity of the implicit relations in the solution process. Advantages and disadvantages of the different approaches are discussed. We compare the methods with respect to efficiency and accuracy of the solution. A Strang-type splitting method combined with approximate matrix factorization is found to be most efficient in the low to modest accuracy range and this range is of interest for the model.