Summary: Various numerical techniques are developed and compared for solving the one-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed by

*R. F. Warming* and

*B. J. Hyett* [J. Comput. Phys. 14, 159–179 (1974;

Zbl 0291.65023)]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The new methods are more accurate and are more efficient than the conventional techniques. These schemes are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared.