Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions

Solutions of conservation laws satisfy the monotonicity property: the numbe r of local extrema is a non-increasing function of time, and local maximum/ minimum values decrease/increase monotonically in time. This paper investig ates this property from a numerical standpoint. We introduce a class of ful ly discrete in space and time, high order accurate, difference schemes, cal led generalized monotone schemes. Convergence toward the entropy solution i s 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 c onvex. We define discrete paths of extrema by tracking local extremum value s in the approximate solution. In the course of the analysis Ne establish t he pointwise convergence of the trace of the solution along a path of extre mum. As a corollary, we obtain a proof of convergence fdr a MUSCL-type sche me that is second order accurate away from sonic points and extrema.