Compact Difference Scheme For Elliptic Equations With Variable Coefficients

Many mathematic models of physics and engineering practical problem can be described by elliptic partial differential equations, such as the diffusion problem, the current distribution problems in conductor and electrostatic problem. But the exact solutions of elliptic equation boundary value problem can be obtained only in exceptional condition. It is essential to solve these problems by numerical method. The finite difference method is a popular and effective numerical method for solving elliptic equation, a number of elliptic equation difference approximation schemes have been constructed up to now. The difference methods for solving one-dimensional elliptic equation are direct difference method, integral interpolation method, variation difference method and so on, but these methods can only reach second order accuracy, in order to improve the accuracy, we need to use fine grid, but this will increase the computation quantity greatly. The accuracy order of many difference schemes for solving two-dimensional elliptic equation with variable coefficients is relatively low. Therefore, constructing a high-accuracy difference scheme for solving elliptic equation with variable coefficients has theoretical and practical significance. The main contents are as follows.Mainly introduce the basic idea of finite difference method and some traditional finite difference schemes for solving elliptic equation with variable coefficients.Construct a fourth-order accuracy compact difference scheme for one-dimensional elliptic equation with variable coefficients, give the prior estimates of the difference solution, and prove the existence, uniqueness, stability and convergence of the difference solution. Then verify the high-accuracy of the difference scheme by numerical experiment.Construct fourth-order accuracy compact difference scheme for two-dimensional elliptic equation with variable coefficients and prove the existence, uniqueness, stability and convergence of the difference solution. Then verify the theoretical result by numerical experiment. Numerical results show that the compact difference scheme in this paper is a high-accuracy, stable and convergence difference scheme.