This dissertation aims to develop various numerical techniques for solving the one dimensional convection–diffusion equation with constant coefficient. These techniques are based on the explicit finite difference approximations using second, third and fourth-order compact difference schemes in space and a first-order explicit scheme in time. The suggested scheme has been seen to be very accurate and a relatively flexible solution approach in solving the contaminant transport equation for Pe ≤ 5. For the solution, the combined technique has been used instead of conventional solution techniques. The accuracy and validity of the numerical model are verified. The computed results showed that the use of the current method in the simulation is very applicable for the solution of the convection-diffusion equation. The technique is seen to be alternative to existing techniques. This dissertation is divided into six chapters: The derivation of the convective diffusion equation is given in Chapter 2. The main idea behind the higher order finite difference technique is given in Chapter 3. The numerical approximations to CDE described with ten different explicit schemes are introduced in Chapter 4. The results of numerical experiments using second, third and fourth-order compact difference schemes are presented in Chapter 5. Chapter 6 is devoted to a brief conclusion. Finally the references are introduced at the end.