Radial basis function collocation method with difference for nonlinear convection-dominated diffusion equations

Abstract

The main purpose of this paper is to introduce a method for solving unsteady nonlinear convection diffusion equations, by incorporating with the difference method, the collocation method and a new radial basis function interpolation. This method overcomes the difficulty that the collocation method cannot be directly applied to nonlinear convection diffusion equations. Further, the existence and uniqueness of the solution to the proposed method are established. In particular, the method is applied to the nonlinear convection diffusion problems to demonstrate its appropriateness: Numerical results and the properties of the new radial basis function show the effectiveness of the method in solving convection diffusion problems. Moreover, in contrast to the traditional finite difference method and characteristic finite element method, the method developed in this paper has some attractive advantages in solving convection dominated problems including eliminating the numerical oscillations phenomenon.