Geodesic
Nonconservative scheme with the isentropic condition in rarefaction waves as applied to the compressible Euler equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 11, pp. 1894-1902

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A previously developed second-order accurate quasi-monotone scheme is tested using the Riemann problem with high initial pressure and density ratios. For shock waves, the scheme is conservative, while, in rarefaction waves, the isentropic condition along the trajectory of a Lagrangian particle is used instead of conservativeness in energy. It is shown that the shock front position produced by the scheme has no considerable errors typical of a representative set of conservative quasi-monotone schemes of various orders of accuracy. The numerical accuracy is significantly improved in the case of moving grids with a contact discontinuity explicitly introduced in the form of a grid node. It is shown how the method can be extended to cover the multidimensional case and the presence of additional terms in the original equations. 
@article{ZVMMF_2013_53_11_a10,
     author = {V. I. Gryn' and A. A. Frolova and A. A. Charakhch'yan},
     title = {Nonconservative scheme with the isentropic condition in rarefaction waves as applied to the compressible {Euler} equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1894--1902},
     publisher = {mathdoc},
     volume = {53},
     number = {11},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_11_a10/}
}
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V. I. Gryn'; A. A. Frolova; A. A. Charakhch'yan. Nonconservative scheme with the isentropic condition in rarefaction waves as applied to the compressible Euler equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 11, pp. 1894-1902. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_11_a10/