Total oscillation diminishing property for scalar conservation laws

We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone Φ : R→ R, we have that |Φ(u)|TV (R) is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently on some numerical computations based on TVD schemes. We also show that Total Variation Diminishing finite volume schemes are TOD and that the fully discrete Godunov scheme is TOD.