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

TL;DR

This paper introduces generalized monotone schemes for conservation laws, proving their convergence to entropy solutions by tracking extrema and establishing discrete entropy conditions, with implications for high-order numerical methods.

Contribution

It develops a new class of fully discrete, high-order schemes called generalized monotone schemes and proves their convergence to entropy solutions using a novel extremum path analysis.

Findings

Proven convergence of generalized monotone schemes to entropy solutions.

Established pointwise convergence of extrema traces in approximate solutions.

Demonstrated convergence of a second-order MUSCL scheme away from sonic points and extrema.

Abstract

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.