2026/07/23 by Paola Francesca Antonietti, Francesca Bonizzoni, Mattia Corti +3
Computer Science · Mathematics · #math.NA #cs.NA
The Fisher-Kolmogorov model is one of the most widely used models in the study of neurodegenerative diseases, owing to its simple structure as a nonlinear reaction-diffusion equation. In particular, it is commonly employed to describe proteinopathies such as Alzheimer's and Parkinson's diseases. Under suitable assumptions, non-negativity of the solution is guaranteed at the continuous level, which is physically relevant since the solution represents a relative concentration. However, this property is not generally preserved at the discrete level, potentially leading to unphysical and unstable numerical approximations. In this work, we analyze a modified version of the Fisher-Kolmogorov model that stabilizes the dynamics around the unstable equilibrium c=0. For the spatial discretization, we adopt a discontinuous Galerkin method on polygonal and polyhedral meshes, coupled with the Crank-Nicolson scheme for time integration. We derive stability and a-priori error estimates for the semi-discrete problem. The theoretical findings are supported by numerical experiments, including convergence studies in both two and three dimensions. Finally, we validate the model through simulations of α-synuclein diffusion in a two-dimensional agglomerated brain section, demonstrating the high-order accuracy and robustness of the proposed method.