Mimetic Properties of Difference Operators: Product and Chain Rules as for Functions of Bounded Variation and Entropy Stability of Second Derivatives

TL;DR

This paper explores mimetic properties of difference operators, demonstrating that second order discretizations can mimic product, chain rules, and entropy dissipation properties of functions of bounded variation, unlike higher order methods.

Contribution

It introduces discrete analogues of product and chain rules for functions of bounded variation and compares the mimetic properties of second order versus higher order difference operators.

Findings

Second order operators satisfy discrete product and chain rules.

Higher order approximations do not generally satisfy these mimetic properties.

Second derivatives with varying coefficients exhibit strong entropy dissipation.

Abstract

For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear systems in several space dimensions is an open problem, mimetic properties such as summation-by-parts as discrete analogue of integration-by-parts allow a direct transfer of some results and their proofs, e.g. stability for linear systems. In this article, discrete analogues of the generalised product and chain rules that apply to functions of bounded variation are considered. It is shown that such analogues hold for certain second order operators but are not possible for higher order approximations. Furthermore, entropy dissipation by second derivatives with varying coefficients is investigated, showing again the far stronger mimetic properties of second order approximations compared to higher order ones.