Fast Numerical Method For A Class Of Three-Dimensional Nonlinear Diffusion Equations

Nonlinear diffusion equations have a wide range of applications in nonlinear science.These equations describe phenomena such as nonlinear diffusion,solute permeation theory,phase transition principle theory,biochemistry,and biological population dynamics.In most cases,nonlinear diffusion equations have serious degradation or other nonlinear characteristics.Researchers have successively proposed efficient numerical methods from different perspectives.This paper mainly studies efficient and fast numerical methods for solving three types of nonlinear diffusion equations:Firstly,the second order central difference scheme is given for the twodimensional semi linear parabolic equation with Dirichlet boundary condition,and the differential matrix of the two-dimensional Laplace operator is written by Kronecker product;Then,the Crank-Nicolson method is used to discretize the time,and Picard iteration is used to solve the discretization to obtain the nonlinear algebraic equations;Finally,in order to improve the computational efficiency,the fast discrete sine transform is used to solve the problem.Secondly,for the d dimensional (d =1,2,3) Poisson equation with Dirichlet boundary conditions,a fast solution method is developed.Firstly,the finite difference method is used to discretize the equation,and the matrix decomposition of the discretized equation is performed by using the properties of Kronecker product;Then the fast discrete sine transform(DST)method is applied to solve the problem effectively;Finally,the calculation speed and error of the two calculation methods using the fast discrete sine transform(DST)method and not using the fast discrete sine transform(DST)method are compared in three cases.Finally,a fast Fourier transform(FFT)numerical method is constructed to solve the three-dimensional nonlinear reaction diffusion equation under the homogeneous Neumann boundary condition.First,the second order central difference scheme is given by using the finite difference method;Then,the differential matrix of the threedimensional Laplace operator is diagonalized,and the Crank-Nicolson method is used for time discretization;Finally,the fast Fourier transform(FFT)is combined to quickly solve the nonlinear algebraic equations.The numerical experiments show that this method can still reduce the memory and CPU computing time significantly while maintaining high accuracy and high dimension.In the first chapter,this paper first introduces the research background,namely the basic concepts and knowledge summary of nonlinear diffusion equation,then describes the research status and development at home and abroad,and finally puts forward the practical value,characteristics and innovation of this study.In the second chapter,the derivation process of Laplace differential matrix is given,and the concepts and properties involved in the following articles are given;In chapter 3,the numerical method of fast sine discrete transformation for semi linear parabolic equation is studied;In chapter 4,the finite difference method is used to solve the d-dimensional(d=1,2,3)Poisson equation,and the calculation speed and error of the two methods are compared in three cases;In chapter 5,the finite difference method based on Fast Fourier Transform(FFT)is studied to solve the three-dimensional reaction-diffusion equation,and the numerical simulation of three-dimensional nonlinear reaction-diffusion equation is further studied.