Mathematical modeling and numerical investigation of traveling waves in biomedicine and geophysics

The main purpose of the thesis consists in performing a numerical investigation of the traveling waves phenomenon by means of some biomedical and geophysical models. As far as the biomedical environment, reaction-diffusion systems for cancer research are examined: suitable Implicit-Explicit strategies are invoked for the systems discretization, showing effectiveness in terms of both computational efforts and qualitative accuracy. Particular emphasis is given to the wave speed approximation problem, providing a space-averaged estimate: this is an original attempt within the reaction-diffusion systems theory, being borrowed from the conservation laws field and it turns out to be a robust landmark in the current mathematical environment as well. Firstly, the analysis is carried out by considering a prototypical PDE model for phase transitions in tumor growth, namely Epithelial-to-Mesenchymal transition and its reverse process, the Mesenchymal-to-Epithelial transition; afterwards, the Gatenby-Gawlinski model for acid-mediated tumour invasion is deeply analyzed and multidimensional simulations are taken into account with emphasis on some important qualitative aspects strongly linked to experimental observations. Finally, in regards to the geophysical context, the Burridge-Knopoff model for earthquakes is studied, namely a spring-block model defined by a system of ODEs provided with a discontinuous right hand side: numerical simulations are performed by relying on a Predictor-Corrector strategy, with the aim of proving the almost convergence property of the wave speeds.