Discontinuous Hermite Collocation and Diagonally Implicit RK3 for a Brain Tumour Invasion Model

Over the past years mathematical models, based on experimental data from MRI and CT scans, have been well developed to simulate the growth of aggressive forms of malignant brain tumours. The tumour growth model we are considering here, apart from proliferation and diffusion, is being characterized by a discontinuous diffusion coefficient to incorporate the heterogeneity of the brain tissue. For its numerical treatment by high order methods, we have developed a Discontinuous Hermite Collocation (DHC) finite element method, with appropriately discontinuous basis functions associated with the discontinuity nodes, to discretize in space. In this work, together with the classical backward Euler and Crank Nicolson schemes, we also consider the deployment of a third order diagonally-implicit Runge-Kutta (RK3) scheme to discretize in time. Several numerical experiments are included to demonstrate the performance of the method. The numerical investigation conducted, reveals that the DHC-RK3 is an order O(τ + h) scheme.