Some Studies On The Finite Difference Methods For The Special Kind Of Convection Diffusion Equation

Convection-diffusion equations are the linearized model of the nonlin- ear equations coming from the viscous fluid dynamics. In addition, they are of their own value in describing many natural phenomena, such as, the diffusion of polluted substances in water or air, the diffusion of heat and salinity in the ocean, etc. So, the computation of the convection-diffusion equations has drawn much attention of many researchers. Consider the initial-boundary convection diffusion equation in the do- main D {.r~0 < ?< 1 0 < I 0 (1) with the initial condition u(x,0) ~uo(x) 0< x <1 and the boundary conditions where s > 0 is an arbitrary constant. In this paper, we constructed some finite difference methods in solv- ing the linear convection diffusion equation from three aspects: the gen- eral difference scheme, the parallel solution scheme and the characteristic- difference scheme. Particularly, we considered the cases when the coeffi- cient of the convection term is infinite while the coefficient of the diffusion term approaches zero. At present, many methods and techniques for the numerical solution of the constant coefficient linear convection-diffusion equations have been 3 LI studied extensively, and the theories are comparatively mature. These methods and techniques can be generalized to solving nonlinear convection- diffusion equations, and the key problem is how to deal with the convection term and the cases when ~ is small. Previous works are focused on the cases when the convection term is finite, in this paper, we consider the case when tins tern is iiifiuiite. In the field of Difference Methods, we obtain two kinds of weighted im- plicit schemes which are absolutely stable, three kinds of weighted explicit schemes which are conditionally stable. In the field of Parallel Methods, we generalize the GEL method, the GER method, the AGE method which are suitable for solving problem(1), and the stable conditions are obtained. In the field of Characteristic-Difference Methods, we obtain two kinds of schemes, the second of which is of small truncation error. Finally, the numerical experiment is made for the following problem: I 8u _ 28u ?a2u u(x,O>1梮2 OO The results show the feasibility of the above methods.