Research On Dynamically Consistent Numerical Methods Of Two Classes Of Reaction Diiffusion Equations

Reaction-diffusion equation is a kind of parabolic partial differential equation,which is often used to study some complex life phenomena quantitatively or qualitatively.In general,it is difficult to find an exact solution to the reaction-diffusion equation.In recent years,people have constructed various numerical methods to find approximate solutions of such equations.Generally speaking,a numerical method has practical significance only if it can reflect the dynamic properties of the continuity equation.Therefore,constructing a numerical method that can maintain the dynamic properties of the reaction-diffusion equation is a subject worthy of study.In this paper,nonstandard finite difference methods for two kinds of reaction-diffusion equations are constructed from the point of keeping the positivity and boundedness of the exact solution of the reaction-diffusion equation as well as the monotonicity of the exact solution in time and space.The second Chapter deals with a class of Burgers-Huxley equations.First,for the space-independent equation of the equation,a non-standard finite difference method that can maintain its dynamic properties is constructed,and then combined with this method,a non-standard finite difference method for solving the Burgers-Huxley equation is proposed.This method can maintain the positivity,boundedness and monotonicity of the exact solution of the continuous equation under certain conditions.In addition,the stability and convergence of the method are also analyzed,and finally some numerical examples are given to verify the correctness of the theoretical results.In Chapter 3,we construct a class of nonstandard finite difference methods to approximate a class of generalized Fisher equations.This equation is a kind of nonlinear reaction diffusion equation,and its solution has dynamic properties such as positiveness,boundedness and monotonicity.First,a class of nonstandard finite difference methods is constructed for spatial independent equations of this kind of equation.Combined with the nonstandard finite difference method of the spatial independent equation,the nonstandard finite difference method of the generalized Fisher equation is constructed.It is proved that the proposed nonstandard finite difference method can maintain the positivity,boundedness and monotonicity of the solution of the equation when the time step denominator function satisfies certain conditions.The stability and convergence of the proposed nonstandard finite difference method are discussed.Finally,the correctness of the theoretical results is verified by numerical simulation.