The Characteristic Difference Method For Solving Convection-Diffu-sion Equations

The convection-diffusion equations are a kind of basic equations, which can be used to describe the distribution of pollutants in the water or in the air, the flow of liquid and the heat exchange of liquid, etc. So it is very significant in theory or in practice to study the numerical algorithm for solving convection-diffusion equations and to seek a quick, practical and stable numerical method. The convection-diffusion equations and their numerical methods are studied systemically in this paper. In order to eliminate the numerical oscillations of the convection-dominated diffusion equation and improve the computational efficiency of the characteristic difference method, several new characteristic difference schemes, based on linear or bilinear interpolation, are constructed in this paper for solving one-dimensional and two-dimensional, linear or nonlinear convection-diffusion equations. Some good results are obtained as follows:1. To the one-dimensional and linear convection-diffusion equation, two kinds of characteristic difference scheme, based on linear interpolation, are obtained and the error analysis, as well as the analysis of convergence is presented. The numerical example shows that the new characteristic difference scheme is better than the previous method in reducing the numerical oscillations of the convection-dominated diffusion equations.2. The one-dimensional and nonlinear convection-diffusion equation is discussed in this paper. Also, two kinds of characteristic difference scheme are formulated, based on linear interpolation. And a one-dimensional and quasilinear convection-dominated diffusion equation is presented. The result shows that the modified algorithm is more effective and accurate.3. The two-dimensional and linear convection-diffusion equation is studied, and three kinds of characteristic difference scheme are created based on bilinear interpolation. A numerical example is presented, too.4. And to the two-dimensional and nonlinear convection-diffusion equation, three kinds of characteristic difference scheme are created respectively, based on bilinear interpolation.To these modified algorithms, it is very easy to select the interpolation points and reduce the numerical errors. And the computational efficiency gets greatly improved with these new algorithms.