Some Studies On Difference Schemes For The Two-Dimensional Parabolic Equations With Neumann Boundary Conditions

Parabolic partial differential equations have the widespread applications in physics,mechanics and engineering technology.The study of numerical methods for the solution of parabolic partial differential equation has a very important role.The difference method for the parabolic partial differential equations with Neumann boundary conditions is a topic of the research.For one-dimensional problems,there are rich results on the differ-ence schemes.This paper is devoted to the two-dimensional parabolic partial differential equation with Neumann boundary conditions.The main content of this paper is divided into two parts.The first part is the construction and maximum norm estimate of the second-order scheme,the second part is the construction maximum norm estimate of the fourth-order compact scheme.The methods of deriving the difference schemes at the inner points are the same as that for the problem with Dirichlet boundary condition.Consider-ing the differential equation at the boundary points and using the boundary condition,we obtain a partial second order scheme with truncation error of O(?2+h12-h22)+ and a fourth order scheme with truncation error of O(?2+h14+h24).This method avoids introducing the fictitious points near the boundary.The existence and uniqueness of the solution and stability of the difference schemes are proved by the discrete energy analysis method.The convergence of the difference schemes in the maximum norm are proved by the H2 estimates together with the discrete embedding theorem.Finally,several examples are given to demonstrate the numerical accuracy of obtained difference schemes.