A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition (Communications in Computational Physics, 2023)
The numerical solutions of gas dynamics equations have to be consistentwith 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 schemewith the hybridized flux developed by combining the acoustic approximate (AA) fluxand 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 inthe part zones of rarefaction waves. Moreover, the EC flux may result in nonphysicalnumerical oscillations in simulating strong rarefaction waves. Another disadvantageof this scheme is that it is of only first-order accuracy. In this paper, we firstly constructa modified entropy conservative (MEC) flux which can damp effectively numerical oscillations in simulating strong rarefaction waves. Then we design a new hybridizedCL scheme satisfying the entropy condition for one-dimensional complex flows. Thisnew 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 beingunder-entropic, we propose using the third-order TVD-type Runge-Kutta time discretization method. Based on the new hybridized flux, we develop the second-orderCL scheme that satisfies the entropy condition.Finally, the characteristics of our new CL scheme using the improved hybridizedflux are demonstrated through several numerical examples.