Tightly Difference Scheme, Convection Diffusion Equation

Convection-diffusion equations are fundamental equations for dynamics which occupy an important position in partial differential equations, and are the linearized model of the nonlinear equations coming from the viscous fluid dynamics. In this paper, based on some of the finite difference theory point of view, theories and methods, the high accuracy compact difference schemes for one and two dimensional convection-diffusion equations are proposed respectively.For one-dimensional convection-diffusion equation, a fourth and sixth order compact difference scheme is derived by studying the relation between the difference operator and the differential operator. And the error analysis for the compact difference scheme is also given. The numerical experiments prove those schemes'effective. Then the spectrum and condition number of tri-diagonal Toeplitz matrix is analyzed separately. Finally,the Richardson and Chen-Lin expand formula is used to compute a sixth order numerical solution of the equation. The numerical experiments show this method has a good effort.For two-dimensional convection diffusion equation, according to the second and fourth compact difference scheme of one-dimensional equation, the five-point and nine-point difference scheme is derived by studying the relation between the difference operator and the differential operator. Then the algorithms, such as, chasing method of sine transform, BICG and BICGSTAB, are used to the numerical solution of equation according to the calculation templates of the compact difference schemes. The numerical experiments prove those schemes'effective.