Geodesic
An implicit finite-difference scheme for the unsteady motion of a viscous barotropic gas
Numerical methods and programming, Tome 14 (2013) no. 4, pp. 516-523.

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A new implicit finite-difference scheme for the unsteady motion of a viscous barotropic gas is proposed in terms of Eulerian coordinates for the cases of one, two, and three spatial variables. An existence and uniqueness theorem for a difference solution of this scheme is proved without any assumptions on grid steps. An important property of the scheme is the fact that the grid density function is always positive. At each time step, a grid solution is found by solving a linear system. A theoretical error estimate for this method is formulated in the case of an exact smooth solution. An efficiency of the proposed scheme is confirmed by solving a problem with discontinuous initial data in the one-dimensional case and by solving a cavity problem in the two-dimensional case.
Keywords: finite-difference scheme; error of numerical solution; viscous gas. 
@article{VMP_2013_14_4_a9,
     author = {A. V. Popov and K. A. Zhukov},
     title = {An implicit finite-difference scheme for the unsteady motion of a viscous barotropic gas},
     journal = {Numerical methods and programming},
     pages = {516--523},
     publisher = {mathdoc},
     volume = {14},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2013_14_4_a9/}
}
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A. V. Popov; K. A. Zhukov. An implicit finite-difference scheme for the unsteady motion of a viscous barotropic gas. Numerical methods and programming, Tome 14 (2013) no. 4, pp. 516-523. http://geodesic.mathdoc.fr/item/VMP_2013_14_4_a9/