A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition

The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics, which is termed entropy condition. How-ever, 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 hy-bridized 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 os-cillations 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.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 dis-cretization method. Based on the new hybridized flux, we develop the second-order CL scheme that satisfies the entropy condition.Finally, the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.