TY - JOUR
T1 - A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition
AU - Li , Zhong-Ze
AU - Liu , Li
AU - Cheng , Jun-Bo
JO - Communications in Computational Physics
VL - 2
SP - 452
EP - 476
PY - 2023
DA - 2023/03
SN - 33
DO - http://doi.org/10.4208/cicp.OA-2022-0092
UR - https://global-sci.org/intro/article_detail/cicp/21496.html
KW - Cell-centered Lagrangian scheme, entropy conditions, modified entropy conservative flux, second-order scheme.
AB - <p style="text-align: justify;">The numerical solutions of gas dynamics equations have to be consistent
with the second law of thermodynamics, which is termed entropy condition. However, most cell-centered Lagrangian (CL) schemes do not satisfy the entropy condition.
Until 2020, for one-dimensional gas dynamics equations, the first-order CL scheme
with the hybridized flux developed by combining the acoustic approximate (AA) flux
and the entropy conservative (EC) flux developed by Maire et al. was used. This hybridized CL scheme satisfies the entropy condition; however, it is under-entropic in
the part zones of rarefaction waves. Moreover, the EC flux may result in nonphysical
numerical oscillations in simulating strong rarefaction waves. Another disadvantage
of this scheme is that it is of only first-order accuracy. In this paper, we firstly construct
a modified entropy conservative (MEC) flux which can damp effectively numerical oscillations in simulating strong rarefaction waves. Then we design a new hybridized
CL scheme satisfying the entropy condition for one-dimensional complex flows. This
new hybridized CL scheme is a combination of the AA flux and the MEC flux.<br/>In order to prevent the specific entropy of the hybridized CL scheme from being
under-entropic, we propose using the third-order TVD-type Runge-Kutta time discretization method. Based on the new hybridized flux, we develop the second-order
CL scheme that satisfies the entropy condition.<br/>Finally, the characteristics of our new CL scheme using the improved hybridized
flux are demonstrated through several numerical examples.</p>