Research And Simulation Of The Behavior Of The Difference Scheme Of A Non-homogeneous Parabolic Equation With Variable Cofficients

Among partial differential equations,parabolic equations are a kind of partial differential equations with related physical background.Parabolic partial differential equations have a wide range of applications in the study of heat conduction processes,partial diffusion phenomena and electromagnetic field transmission,and many other problems,and have more extensive application prospects.Over the years,there have been many results in the research of parabolic equations with constant coefficients.In the field of engineering technology,especially in the fields of geophysics and materials science,more and more attention has been paid.The study of partial differential equations of degree is somewhat difficult.Because of the wide range of applications in practical problems,it is of great practical significance to use numerical methods to find its approximate solutions.Variable coefficient non-homogeneous parabolic equations include There are many types,which can be variable coefficients about space and time respectively;or they can have variable coefficients in space and time.When studying variable coefficient parabolic equations,the free term of the equation itself and the nature of the variable coefficients lead to the correct solution.The initial value is very sensitive,and the numerical calculation results are not easy to obtain.Even if a numerical equation with a small amount of calculation and a fast calculation speed can be established,we must pay more attention to the solvability,stability and convergence of the numerical solution.The problem of concern can be fully proved theoretically.The finite difference method is one of the simple,efficient,and widely used methods for solving parabolic equations.This article will use the finite difference method to establish a high-precision second-order difference scheme for parabolic equations with variable coefficients and inhomogeneous;and strictly analyze the solvability,stability and convergence of the difference scheme.It is divided into four parts to explain it.The specific content plan is as follows.The first part mainly explains the research background and significance of non-homogeneous one-dimensional parabolic equations with variable coefficients,as well as the progress of related numerical methods in the research of onedimensional parabolic equations for many years,and elaborates the main research content and significance of this article;In the second part,a second-order difference scheme of a non-homogeneous one-dimensional parabolic equation with variable coefficients is first established,and then some lemmas are listed,that is,the priori estimate of the solution of the difference equation;then the discrete energy method is used Strictly analyze the solvability,stability and convergence of the solution of the difference scheme.At this time,the time and space accuracy are both second-order.In order to improve the accuracy,the truncation error equation is analyzed carefully to discretize it,and the Richardson extrapolation algorithm is used to make the numerical accuracy is increasing to fourth-order in space and time,which is also the biggest highlight of this article.This method not only makes the numerical solution with high precision,but also is unconditionally stable,that is,it can take any time and space step.Finally,the numerical value is given.The calculation example is used to verify the theoretical proof part through data simulation.The data simulation result fully demonstrates the correctness of the theoretical proof part and the calculation is efficient;the third part mainly wants to extend the above scheme to a two-dimensional parabola with variable coefficients and inhomogeneous equations are explained in this part;the fourth part summarizes the whole paper,expounds the results of this paper,and looks forward to the work that can be done in the future on this basis.