一种用于Fisher-Kolmogorov模型的保持稳定性的多面体间断Galerkin方法及其在神经退行性疾病中的应用
A stability-preserving polytopal discontinuous Galerkin method for the Fisher-Kolmogorov model with applications to neurodegenerative disease modelling
AI总结：
研究针对神经退行性疾病中广泛使用的Fisher-Kolmogorov模型，在离散层面解非负性不保的问题，采用多边形和多面体网格上的间断Galerkin方法及Crank-Nicolson格式，推导稳定性和误差估计，经实验验证方法具高阶精度和鲁棒性。
AI中文摘要：
Fisher-Kolmogorov模型是神经退行性疾病研究中广泛使用的模型，因其结构简单，常用于描述如阿尔茨海默病和帕金森病等蛋白质病。在合适假设下，连续层面解非负，但离散层面该性质通常不保，可能导致非物理和不稳定数值近似。本文分析修正的Fisher-Kolmogorov模型以稳定不稳定平衡态c = 0周围动力学。空间离散采用多边形和多面体网格上的间断Galerkin方法，时间积分用Crank-Nicolson格式。推导了半离散问题的稳定性和先验误差估计。数值实验支持了理论结果，最后通过二维凝聚脑切片中α-突触核蛋白扩散模拟验证了模型，证明了所提方法的高阶精度和鲁棒性。
英文摘要：
The Fisher--Kolmogorov equation models the spatio-temporal evolution of interacting biological species and is extensively employed in fields such as ecology, population dynamics, and the modelling of neurodegenerative diseases. Under suitable assumptions on the data, the solution $c$ is non-negative, a key feature because $c$ typically denotes a population density or a relative concentration. However, standard discretisation methods often fail to preserve this property, resulting in non-physical oscillations and unstable numerical approximations. In this work, we propose and analyse a numerical method to stabilise the dynamics of the Fisher--Kolmogorov model around the unstable equilibrium $c=0$. The proposed approach combines a discontinuous Galerkin spatial discretisation on general polygonal and polyhedral meshes with the Crank--Nicolson time integration scheme. The main idea is to suitably modify the formulation at the continuous level so that, on the one hand, it is strongly consistent with the original model, and, on the other hand, it ensures stability when moving to the discrete setting. We prove well-posedness of the semi-discrete formulation, derive stability bounds and prove optimal \textit{a priori} error estimates in a suitable energy norm. The theoretical results are demonstrated through a comprehensive set of numerical examples. Moreover, we consider an application arising in computational neuroscience by simulating the propagation of $α$-synuclein, a key pathogenic protein implicated in Parkinson's disease and other neurodegenerative diseases, demonstrating that the proposed scheme is stable, high-order accurate, and robust in a biologically relevant computational setting.
