Entropy production in second-order three-point schemes

Abstract

We discuss semi-discrete three-point finite difference methods for the numerical solution of system of conservation laws which are second order accurate in space in the sense of truncation error. Particular discretizations of the numerical entropy flux associated with such schemes are studied clarifying the importance of this discretization with regard to the production of numerical entropy. Using a numerical entropy flux constructed in a canonical way we prove that a wide class of finite difference methods cannot satisfy a discrete entropy inequality. Together with a well known result of Schonbek concerning Lax-Wendroff type schemes our result indicates a strong relationship between entropy production and oscillations in numerical solutions.