Numerical Entropy Production on Shocks and Smooth Transitions

In this work, we describe the behaviour of the numerical cell entropy production for several schemes. The numerical results we show indicate that numerical entropy production can be used to estimate the local error in regions of smoothness and to locate shocks. Thus the numerical entropy production can be computed at each time step to monitor the numerical solution produced by a scheme. The information gained in this fashion can be used in strategies of grid adaptivity and/or to enhance the resolution of discontinuities. We also describe how, at least for central schemes, the evaluation of numerical entropy production does not result in a substantial extra computational cost. We consider the one-dimensional system of conservation laws: