Research On The Finite Difference Methods For The Nonlinear Partial Differential Equations

Nonlinear partial differential equation can describe many phenomena in the objective physical world,and more and more people begin to study it in recent years.In this paper,the common numerical solutions of several kinds of nonlinear partial differential equations are studied,including heat conduction equation with nonlinear convection term,two-dimensional diffusion equation and KDV-Burgers equation.According to the characteristics of the equation,finite difference method and non-standard finite difference method are mainly used in numerical method.Firstly,the heat conduction equation with nonlinear convection term is introduced,and the finite difference method is used to solve the equation numerically.Combined with a specific numerical example,numerical experiment is carried out,and the numerical solution is compared with the real solution,and the error is analyzed.Finally,the conditions for the effectiveness and feasibility of the method are discussed.Secondly,a new high-precision two-layer explicit difference scheme is established for the two-dimensional reaction-diffusion equation.Firstly,the difference scheme is established,and then the boundary points of the second derivative term are discretized by Taylor expansion and central difference scheme,and the difference scheme is obtained after sorting.Then,the Fourier method was applied to analyze the stability of the difference scheme,and the conditions satisfying the stability of the difference scheme were obtained.Finally,numerical experiments were carried out for specific examples,and the numerical solutions were compared with the real ones by using Matlab software.The results show that the scheme is feasible and stable under certain conditions.Finally,the nonstandard finite difference method is used to numerically solve the KDV-Burgers equation,and the stability of the method is proved.Through numerical experiments with specific examples,it is found that the absolute error obtained by the nonstandard finite difference method is smaller than that obtained by the standard finite difference method,and the solution obtained is more accurate.The stability of the numerical results is also verified.