On the Relation of the Entropy Balance with the Numerical Solution of Systems of Conservation Laws

This work deals with the relation between the numerical solutions of hyperbolic systems of conservation laws and the associated entropy evolution. An analysis of the continuum problem by means of variational calculus clearly emphasizes the consequences of the adopted reconstruction procedure on the induced entropy balance. A methodology is proposed that allows for a posterior local and global spurious entropy production estimates on the basis of an additional equation representing a discrete approximation to the entropy inequality. The problem of defining a consistent approximation of the numerical entropy flux is also addressed in detail. Properly designed numerical experiments support the analysis and contribute to providing a more comprehensive evaluation of the numerical entropy dynamics.