  Solutions to conservation laws satisfy the monotonicity property: the number
of local extrema is a non-increasing function of time, and local
maximum/minimum values decrease/increase monotonically in time. This paper
investigates this property from a numerical standpoint. We introduce a class of
fully discrete in space and time, high order accurate, difference schemes,
called generalized monotone schemes. Convergence toward the entropy solution is
proven via a new technique of proof, assuming that the initial data has a
finite number of extremum values only, and the flux-function is strictly
convex. We define discrete paths of extrema by tracking local extremum values
in the approximate solution. In the course of the analysis we establish the
pointwise convergence of the trace of the solution along a path of extremum. As
a corollary, we obtain a proof of convergence for a MUSCL-type scheme being
second order accurate away from sonic points and extrema.
