Variation entropy: a continuous local generalization of the TVD property using entropy principles

This paper presents the notion of a variation entropy. This concept is an entropy framework for the gradient of the solution of a conservation law instead of on the solution itself. It appears that all semi-norms are admissible variation entropies. This provides insight into the total variation diminishing property and justifies it from entropy principles. The framework allows to derive new local variation diminishing properties in the continuous form. This can facilitate the design of new numerical methods for problems that contain discontinuities. •The notion of variation entropy is presented.•This is an entropy concept based on the gradient of the solution.•It justifies the TVD property from an entropy perspective.•Variation entropies are convex homogeneous functions.•The variation entropy concept serves to facilitate the design of numerical methods which preclude spurious oscillations.