Entropy analysis and improvement of the one-dimensional cell-centered Lagrangian scheme

It is physically admissible that the numerical solutions of gas dynamics equations are consistent with the second law of thermodynamics, but the other cellcentered Lagrangian (CL) schemes except the hybridized CL scheme introduced by Maire et al. do not satisfy this consistency. For one-dimensional gas dynamics equations, this hybridized CL scheme combines the acoustic approximate (AA) flux and the entropy conservative (EC) flux. By distinguishing expanded zones and compressed zones, the hybridized CL scheme basically satisfies the entropy consistency, but in the part zones of rarefaction waves, the hybridized CL scheme is under-entropic and the EC flux may result in the numerical oscillations in simulating strong rarefaction waves. In this paper, we try to design a new hybridized CL scheme satisfying entropy consistency for one-dimensional complex flows. This new hybridized CL scheme is the combination of the AA flux and the modified entropy conservative (MEC) flux. Just like the EC flux, the MEC flux keeps the entropy conservative in simulating rarefaction waves; while differently, the MEC flux damps effectively numerical oscillations in zones of rarefaction waves. And we find that, by using the third-order TVD-type Runge-Kutta time discretization method, the CL scheme with the improved hybridized flux can completely satisfy entropy consistency. Finally, some numerical examples are tested to prove the characters of our new CL scheme with the improved hybridized flux.