We consider 3-point numerical schemes for scalar Conservation Laws, that are
oscillatory either to their dispersive or anti-diffusive nature. Oscillations
are responsible for the increase of the Total Variation (TV); a bound on which
is crucial for the stability of the numerical scheme. It has been noticed
(\cite{Arvanitis.2001}, \cite{Arvanitis.2004}, \cite{Sfakianakis.2008}) that
the use of non-uniform adaptively redefined meshes, that take into account the
geometry of the numerical solution itself, is capable of taming oscillations;
hence improving the stability properties of the numerical schemes.

In this work we provide a model for studying the evolution of the extremes
over non-uniform adaptively redefined meshes. Based on this model we prove that
proper mesh reconstruction is able to control the oscillations; we provide
bounds for the Total Variation (TV) of the numerical solution. We moreover
prove under more strict assumptions that the increase of the TV -due to the
oscillatory behaviour of the numerical schemes- decreases with time; hence
proving that the overall scheme is TV Increase-Decreasing (TVI-D).