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.

Publication:

SIAM Journal on Scientific Computing Volume 46, Issue 2, A1132 - A1159: Apr 2024

https://doi.org/10.1137/22M1524151

Author:

YONG LIU

LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, People's Republic of China

Email: yongliu@lsec.cc.ac.cn

JIANFANG LU

School of Mathematics, South China University of Technology, Canton, Guangdong 510641,People's Republic of China

Email: jflu@scut.edu.cn

CHI-WANG SHU

Division of Applied Mathematics, Brown University, Providence, RI 02912 USA

Email: chi-wang shu@brown.edu