Entropy Satisfaction of a Conservative Shock-tracking Method

In this paper we discuss the entropy satisfaction of the conservative shock-tracking technique developed in [D. K. Mao, J. Comput. Phys., 92 (1991), pp. 422–455], [D. K. Mao, J. Comput. Phys., 103 (1992), pp. 359–369], and [D. K. Mao, SIAM J. Numer. Anal., 32 (1995), pp. 1677–1703] when it is applied to the Godunov scheme. We consider the scalar case in one-space dimension and assume that both the flux and entropy functions are strictly convex. We prove that, when the tracked shocks are strong enough in comparison with the variation of the numerical solution around them, the numerical solution satisfies the entropy condition in a certain sense. We also discuss the entropy situation when the convexity of the flux function is very weak and the tracked shock neighbors to strong simple waves.