Research On Embedded Boundary Algorithm In Numerical Simulation Of Alzheimer's Disease Model

Alzheimer's disease(AD)is a degenerative disease of the central nervous system,which brings great distress to people's lives and its pathogenesis is very complicated.Therefore,modeling and analyzing the concentration of substances around neurons can help to explore their pathogenesis.This article first introduces the basic model of Alzheimer's disease,The substance f is assumed to be involved in the growth process of g and in contrast,the substance g inhibits the production of f and finds that the difficulty of solving this model is the solution of the Laplace operator,and the basic idea of embedded boundary algorithm used to solve the Alzheimer's disease model is the finite volume method.In order to further deepen the understanding of the problem,from simple to difficult to analyze the problem,first take the Poisson equation containing Laplace operator as an example,use the finite volume method to solve the Poisson equation,and the analysis results in the Poisson equation One-dimensional and two-dimensional discrete formats,using quadratic interpolation method to solve the unknown cells,using MATLAB to obtain the results of one-dimensional two-dimensional numerical simulation,and calculating the second norm of the errors of different grid points,the convergence order of the method is second order.Theoretically,the compatibility of flux is studied by Taylor expansion,and the compatibility of flux is second order.Analyzing the error and finding that the numerical method is second-order convergent.Then,using embedded boundary algorithm,for different boundary problems,the processing methods of different cutting cells are analyzed under the one-dimensional and two-dimensional boundary conditions of the Poisson equation,and the boundary unknown cells.The estimated value is obtained by numerical simulation,and the result of one-dimensional two-dimensional numerical simulation is calculated.The second norm of the error of different grid points is calculated.The convergence order of the method is second-order.In the case of different positions,it is found that its position has nothing to do with the size of the error.Theoretically,the compatibility of flux is studied by Taylor expansion,and the compatibility of flux is second order.Analyzing the error and finding that the numerical method is second-order convergent.Finally,the iterative form of the Crank-Nicolson method using the fixed point strategy gives the numerical solution algorithm of Alzheimer's disease,and embedded boundary algorithm is used to numerically simulate the Alzheimer's disease model.The two cases of different diffusion coefficients are given:when the diffusion coefficient is small,the substance f that promotes neuron communication will eventually disappear.The activity of neurons will become very low,the communication between neurons will stagnate,aggravating the development of Alzheimer's disease,when the diffusion coefficient is large,when the substance f that promotes neuron communication and the substance that hinders substance g will be in a relatively balanced state,the activity of neurons will be high,and communication between neurons will be normal.