New Finite Difference Schemes For Convection-Diffusion Equations

The convection-diffusion equation with small singularly perturbed parameter can be found in many fields of physics and engineering. Such as the Navier-Stokes equation with high Reynolds numbers in the fluid. However, there are some difficulties in constructing efficient numerical methods for this kind of equations for the existence of the so-called boundary layer. And the well-known nonphysical oscillation will occur when classical finite difference method or finite element method are used. So, to overcome these difficulties and construct robust and efficient numerical methods which can be used to obtain precise numerical solutions are useful and meaningful.In this paper, we mainly consider a kind of new finite difference(NFD) methods for solving the convection-diffusion equation with small singularly perturbed parameters in one and two dimensions. Assuming that the mesh size h and the singularly perturbed parameter ? satisfying the condition h/? = C(C is a constant independent of ? and h), using the stability of the solution and the triangle function theorem, we construct a series of NFD schemes for the one-dimensional problems with constant and variable coefficients firstly in the second section and third section respectively. And the error estimate for problems with constant coefficient is proved as well. Then, applying the ADI technique, the idea is extended to the equations in two dimensions directly in the fourth section. And related numerical experiments are also shown to illustrate the effectiveness of the above proposed schemes. There are mainly two advantages for the proposed methods: one is that the NFD schemes can achieve the predicted convergence orders regardless of the perturbed parameter; the other is, no matter which convergence order the new scheme is, the generated linear systems have tri-diagonal and nine-diagonal structure for the one and two dimensional problems respectively. Therefore, the new schemes are very efficient and easy to implement for the singularly perturbed problem.