Convection-diffusion Equation Finite Difference Method

A class of convection-diffusion equation is the basic equation of motion, it can be used to describe the river pollution, air pollution, nuclear waste pollution in the distribution of contaminants, the fluid flow and heat transfer fluids, and many other physical phenomena. The convection-diffusion partial differential equations (or equations) have a common characteristic:the impact of convection than diffusion, the convection dominance. Convection of a numerical solution of convection-diffusion problems brought many difficulties, so convection effective numerical solution of the problem has always been important research in computational mathematics content, with the usual difference method or finite element method appears to solve numerical oscillations.Based on the finite difference solution of partial differential equations and the characteristic finite difference method.First of all, the background of the convection-diffusion equation, and then introduced the current method convection-diffusion equation, and then use compact finite difference method, the characteristic finite difference method in depth the two methods of constant coefficient convection-diffusion equation with variable coefficient finite difference scheme. Compact finite difference method of high precision, the fourth-order compact scheme will not need step by step process of convection and diffusion considerations, and involves only adjacent grid points, can obtain very accurate coarse grid results. Consider the characteristic finite difference method along the characteristic line (flow direction) of the discrete convection-diffusion problems using the physical and mechanical characteristics, can effectively overcome the numerical oscillations, to ensure the stability of numerical solutions.