Numerical methods for advection-diffusion-reaction equations and medical applications

The purpose of this thesis is twofold, firstly, the study of a relaxation procedure for numerically solving advection-diffusion-reaction equations, and secondly, a medical application. Concerning the first topic, we extend the applicability of the Cattaneo relaxation approach to reformulate time-dependent advection-diffusion-reaction equations, that may include stiff reactive terms, as hyperbolic balance laws with stiff source terms. The resulting systems of hyperbolic balance laws are solved by extending the applicability of existing high-order ADER schemes, including well-balanced and non-conservative schemes. Moreover, we also present a new locally implicit version of the ADER method to solve general hyperbolic balance laws with stiff source terms. The relaxation procedure depends on the choice of a relaxation parameter $\epsilon$. Here we propose a criterion for selecting $\epsilon$ in an optimal manner, relating the order of accuracy $r$ of the numerical scheme used, the mesh size $\Delta x$ and the chosen $\epsilon$. This results in considerably more efficient schemes than some methods with the parabolic restriction reported in the current literature. The resulting present methodology is validated by applying it to a blood flow model for a network of viscoelastic vessels, for which experimental and numerical results are available. Convergence-rates assessment for some selected second-order model equations, is carried out, which also validates the applicability of the criterion to choose the relaxation parameter. The second topic of this thesis concerns the numerical study of the haemodynamics impact of stenoses in the internal jugular veins. This is motivated by the recent discovery of a range of extra cranial venous anomalies, termed Chronic CerbroSpinal Venous Insufficiency (CCSVI) syndrome, and its potential link to neurodegenerative diseases, such as Multiple Sclerosis. The study considers patient specific anatomical configurations obtained from MRI data. Computational results are compared with measured data.