When solving convection-diffusion equations using the finite difference schemes, the convection term is usually discretized by the upwind schemes to avoid oscillations. A method to eliminate the convection term from convection-diffusion equations is presented in this paper. The new approach makes it feasible to solve convection-diffusion equations using central difference schemes without oscillations. It can also be easily combined with the Pade approximation to achieve fourth-order accuracy. Numerical examples involving one-dimensional equations are presented in the paper to demonstrate the accuracy and robustness of the new approach.