Mimetic properties of difference operators: product and chain rules as for functions of bounded variation and entropy stability of second derivatives (Q1999719)

The paper discusses the mimetic properties of difference operators applied in the discretisation of hyperbolic conservation laws, and in discrete schemes for PDEs in general. Such schemes used as in continuous operators many laws such as summation-by-parts as discrete analogue of integration-by-parts. In the present paper, the author investigates precisely the discrete analogues of the generalized product and chain rules for functions of bounded variation. Such formulations are shown to hold for some second order operators but not valid for higher order cases. Besides, an entropy dissipation has been proved using second derivatives with varying coefficients. The paper is globally is a good contribution in the whole field of numerical methods for PDEs, and it opens further a nice future about analogue investigation of higher order operators, where the present method could not be applied. What necessary modification(s) should be involved to obtain analogue approximations for higher order cases?