Geodesic
Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1304-1316

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For the quasi-gasdynamic system of equations, there holds the law of nondecreasing entropy. Difference methods based on this system have been successfully used in numerous applications and test gasdynamic computations. In theoretical terms, however, for standard spatial discretizations of this system, the nondecreasing entropy law does not hold exactly even in the one-dimensional case because of the mesh imbalance terms. For the quasi-gasdynamic equations, a new conservative spatial discretization is proposed for which the entropy balance equation has an appropriate form and the entropy production is guaranteed to be nonnegative (which also holds in the presence of body forces and heat sources). An important element of this discretization is that it makes use of nonstandard space-averaging techniques, including a nonlinear “logarithmic” averaging of the density and internal energy. The results hold on arbitrary nonuniform meshes. 
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     author = {A. A. Zlotnik},
     title = {Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1304--1316},
     publisher = {mathdoc},
     volume = {52},
     number = {7},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a12/}
}
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A. A. Zlotnik. Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1304-1316. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a12/