标题： 保结构的LDG离散化Fisher-Kolmogorov方程用于神经退行性疾病建模 显示英文标题

标题： A structure-preserving LDG discretization of the Fisher-Kolmogorov equation for modeling neurodegenerative diseases

摘要： 本文提出了一种保结构、高阶、无条件稳定的数值方法，用于在多边形网格上逼近Fisher-Kolmogorov方程的解，特别关注其在模拟神经退行性疾病中错误折叠蛋白传播的应用。 通过引入熵变量重新构建模型问题，以确保解的正性、有界性和满足离散熵稳定性不等式。该方案结合了多边形网格上的局部间断Galerkin方法进行空间离散化，并使用$\nu$步向后差分公式进行时间积分。 讨论了实现细节，包括牛顿迭代产生的线性系统的详细推导。 通过广泛的数值测试展示了所提方法的精度和鲁棒性。 最后，通过从MRI数据分割出的二维脑几何形状中模拟\textalpha{}-突触核蛋白传播的仿真，展示了该方法的实际性能，为建模突触病（如帕金森病）以及更普遍的神经退行性疾病提供了一个相关的计算框架。 显示英文摘要

摘要： This work presents a structure-preserving, high-order, unconditionally stable numerical method for approximating the solution to the Fisher-Kolmogorov equation on polytopic meshes, with a particular focus on its application in simulating misfolded protein spreading in neurodegenerative diseases. The model problem is reformulated using an entropy variable to guarantee solution positivity, boundedness, and satisfaction of a discrete entropy-stability inequality at the numerical level. The scheme combines a local discontinuous Galerkin method on polytopal meshes for the space discretization with a $\nu$-step backward differentiation formula for the time integration. Implementation details are discussed, including a detailed derivation of the linear systems arising from Newton's iteration. The accuracy and robustness of the proposed method are demonstrated through extensive numerical tests. Finally, the method's practical performance is demonstrated through simulations of \textalpha{}-synuclein propagation in a two-dimensional brain geometry segmented from MRI data, providing a relevant computational framework for modeling synucleopathies (such as Parkinson's disease) and, more generally, neurodegenerative diseases.