A new MOX Report entitled “A stability-preserving polytopal discontinuous Galerkin method for the Fisher-Kolmogorov model with applications to neurodegenerative diseases” by Antonietti, P.F.;Bonizzoni, F.;Corti, M.; De March, N.;Di Noto, S.;Regazzoni, F. has appeared in the MOX Report Collection.
Check it out here: https://www.mate.polimi.it/biblioteca/add/qmox/66-2026.pdf
Abstract: 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 alpha-synuclein diffusion in a two-dimensional agglomerated brain section, demonstrating the high-order accuracy and robustness of the proposed method.
