Non-uniform Finite Difference Scheme Method For Convection-diffusion Equation

Convection-diffusion equation is one of the basic equations in the field of fluid mechanics. At the present stage,there are still some difficulties to search the exact solution of most part of such equations,So the studies of numerical solution become a focus of attention. The common numerical methods includes:finite difference method, finite volume method and finite element method.Among them, finite difference method is applied more widely for its simple difference scheme and the advantages of easy calculation.The finite difference method, it also involves the meshing, solution of the boundary layer treatment, precision and other key problems.This paper we consider the new compact scheme of convection-diffusion equation, breaking the traditional method of interval subdivision through a central point.By Taylor expansion to construct a finite difference of non-uniform scheme, but also take into account the situation of equidistance. Then intercept the limited entry, but rather take an infinite number (these items constitute a convergent series), trying to make the final difference scheme has a better accuracy.In this paper, in the first section, it is a brief introduction about the source of the problem, the background of convection-diffusion equation as well as the achievements and progress of the latest phase to deal with convection-diffusion equation.In sectionâ…¡, the first part derived in detail to address the problem equidistant difference scheme by an equivalent problem, and gives the format of the error and convergence analysis. The second part of this section for the one-dimensional diffusion equation, starting from the equation itself is constructed of non-equidistant difference scheme, and gives several common non-uniform grid approach. The third part through the numerical experiments show that the finite difference of scheme has better calculation results, and coincides with the analysis of the first part.In sectionâ…¢, We have access to a five-point difference scheme by using the thinking of one-dimensional case, and also compare with the five-point central difference scheme. By numerical example, we reveals the calculation in different situations.