This paper discusses approaches to monotonicity using the advection equation as an example. Linear finite elements are used for spatial approximation. A two-level scheme with weights serves as the time approximation. It is established that the weight of the two-level scheme acts not only as a stabilizer but also as a monotonicity enforcer for the class of problems considered. Various approaches to monotonicity based on the principle of regularization of difference schemes are examined. A symmetric two-level scheme is chosen as the generating scheme. The capabilities of monotonicity are demonstrated on a problem involving the advection equation. The potential of the discussed approaches for nonlinear problems is illustrated with an example of the Burgers’ equation.