Two Numerical Methods For The Two-Dimension Nonlinear Sobolev Equation

The Sobolev equation is widely used in many physical problems, such as:soil moisture migration problem, the permeability theory of fluid throughing the crack of the rock, the heat conduction problem of different medium and so on. So it becomes a eager concern problem what many experts and scholars pay close attention to. In this paper, for the nonlinear Sobolev equation, we use block-centered finite difference method and the char-acteristics of block-centered finite difference method to solve the equation. Not only get the approximate solution of the equation, we can also get the approximate solution of the first order derivative of the solutions, and we can obtain the approximate solution of the first order derivative of the solutions is super convergence.First of all, there is an overview of the background and development status of equation and the method to solve the equation, in addition,simply introduced the work we do and the basic theory of knowledge involved in this paper.Moreover, for the two-dimension nonlinear Sobolev equation with initial boundary value problem, we use difference quotient instead of differential, thus structure the block centered difference format in an isometric mesh, and do its error estimation, the corre-sponding numerical examples were given; To maintain the important physical properties of the equation,the paper combinate the characteristic line and the block-centered difference methods to construct characteristics of block-centered finite difference format, analyses and proves the convergence and stability. Finally, the summary of the equation was given.