An Entropy Stable Essentially Oscillation-Free Discontinuous Galerkin Method for Hyperbolic Conservation Laws

.Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one from among the infinite many weak solutions. Recently, several high order discontinuous Galerkin (DG) methods satisfying entropy inequalities were proposed; see [T. Chen and C.-W. Shu, J. Comput. Phys., 345 (2017), pp. 427–461; J. Chan, J. Comput. Phys., 362 (2018), pp. 346–374; T. Chen and C.-W. Shu, CSIAM Trans. Appl. Math., 1 (2020), pp. 1–52] and the references therein. However, high order numerical methods typically generate spurious oscillations in the presence of shock discontinuities. In this paper, we construct a high order entropy stable essentially oscillation-free DG (OFDG) method for hyperbolic conservation laws. With some suitable modification on the high order damping term introduced in [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299–1324; Y. Liu, J. Lu, and C.-W. Shu, SIAM J. Sci. Comput., 44 (2022), pp. A230–A259], we are able to construct an OFDG scheme with dissipative entropy. It is challenging to make the damping term compatible with the current entropy stable DG framework, that is, the damping term should be dissipative for any given entropy function without compromising high order accuracy. The key ingredient is to utilize the convexity of the entropy function and the orthogonality of the projection. Then the proposed method maintains the same properties of conservation, error estimates, and entropy dissipation as the original entropy stable DG method. Extensive numerical experiments are presented to validate the theoretical findings and the capability of controlling spurious oscillations of the proposed algorithm.Keywordshyperbolic conservation lawsentropy stabilitysummation-by-partsdiscontinuous Galerkin methodnonoscillatoryMSC codes65M1265M6076M20