The Discussion About Some Difference Schemes For Convection-Diffusion Equations

This paper discusses some difference schemes for one dimensional convection-diffusion initial-boundary problems as follows:We begin with the central difference schemes. Combining with the simplest explicit scheme and implicit scheme, the usual Crank-Nicolson difference scheme(six-point implicit scheme) which is denoted to lygs for short, is deducted in Chapter 1. And Chapter 2 is due to present the schemes that can make the coefficient matrix diagonally dominant. Based on the Crank-Nicolson scheme, two kinds of new schemes (penalty scheme, up-wind scheme) which are respectively denoted to fgs and yfgs for short, are presented for variable coefficient problems to make the coefficient matrix diagonally dominant. The new schemes are discussed on cutoff error, precision. In addition, given some limits to the coefficients, convergence analysis is made with respect to L₂ norm.Generally speaking, when central difference methods are used to solve problems, the results are satisfactory. But numerical oscillations often occur when central difference methods are used to solve convection-dominated diffusion problems. The characteristic difference methods can avoid numerical oscillations. In Chapter 3, considered with the idea of interpolation with the closest points, the characteristic difference method based on linear interpolation is improved. The improved method proves simple and convergent with respect to maximum norm.The numerical results tell us the Crank-Nicolson scheme is of highest precision, yfgs lower and fgs lowest. Usually, yfgs is better than fgs because the former is more widely-used and of higher precision. But for the steep front problem which is convection-dominated, numerical oscillations occur when Crank-Nicolson scheme or yfgs is used. While using fgs to solve such problems, numerical oscillations will be avoided. Therefore, fgs still can be an alternative scheme for convection-dominated problem, though a little numerical diffusion occur. As for the improved method presented in Chapter 3, the new characteristic difference method based on linear interpolation is simple and non-oscillatory.