58 ISSN 1064-5624, Doklady Mathematics, 2018, Vol. 97, No. 1, pp. 58–61. © Pleiades Publishing, Ltd., 2018. Original Russian Text © F.B. Imranov, G.M. Kobel’kov, A.G. Sokolov, 2018, published in Doklady Akademii Nauk, 2018, Vol. 478, No. 4, pp. 388–391. Finite Difference Scheme for Barotropic Gas Equations F. B. Imranov a, *, G. M. Kobel’kov b, **, and A. G. Sokolov b, *** Presented by Academician of the RAS E.E. Tyrtyshnikov September 15, 2017 Received September 22, 2017 Abstract—An implicit finite difference scheme approximating the equations of barotropic gas flow is pro- posed. This scheme ensures the positivity of density and the validity of an energy inequality and the mass con- servation law. The continuity equation is approximated implicitly. It is proved that the resulting system of nonlinear equations has a solution for any time and space stepsizes. An iterative method for solving the system of nonlinear equations at each time step is proposed. DOI: 10.1134/S1064562418010179 Constructing finite difference schemes for gas dynamics equations and using these schemes for numerical modelling of compressible ideal (viscous) gas flows is the subject of a huge number of publica- tions (see, for example, [1, 2] and references therein). However, theoretical studies are almost absent. For example, the natural question as to whether a finite difference scheme ensures the positivity of density (which follows from physics) remains open. An excep- tion is the monograph [3], where the positivity of den- sity is proved for the equations of viscous compressible gas in the one-dimensional case, and Zlotnik’s papers (see, for example, [4]), where the underlying system of equations is transformed assuming that the density is positive and a finite difference scheme is constructed for this system, yielding the positivity of density. In the present paper, we propose a finite difference scheme approximating the gas dynamics system of equations that ensures the positivity of density and the conservation of mass balance. The continuity equa- tion is approximated implicitly with respect to the density. Additionally, we prove the existence of a solu- tion to the finite difference problem for any relations between the time and space stepsizes, and we also prove an energy inequality. An iterative process is pro- posed for solving the system of nonlinear equations arising at every time step. 1. STATEMENT OF THE PROBLEM AND A PRIORI ESTIMATES The system of equations describing flows of ideal barotropic compressible gas has the form (see, for example, [5]) (1) For simplicity, below we set . Equations (1) are considered in the cylinder . They are supplemented by the boundary and initial conditions (2) Taking the scalar product of the first equation in (1) (the continuity equation) with and the second equation in (1) with and summing up the results, we obtain After simple transformations and integration with respect to time, we have the “energy identity” γ ∂ρ ∂ρ + = , ∂ ∂ ∂ρ ρ ∂ + + = , ∂ ∂ ∂ = ρ, γ= > . 2 ( ) 0 ( ) ( ) 0 const 1 u t x u u p t x x p a = 1 a = , × , [0 1] [0 ] T Q T , = , = , , = , ρ , =ρ > . 0 0 (0 ) (1 ) 0 ( 0) () ( 0) () 0 u t u t ux u x x x - 2 1 2 u u - ρ, - ρ , + ρ , + ρ , + , = . 2 2 2 1 1 ( ) (( ) ) 2 2 (( ) ) (( ) ) ( ) 0 t x t x x u u u u u u u p u MATHEMATICS a Institute of Physics of National Academy of Sciences of Azerbaijan, AZ1141, Baku, Azerbaijan b Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia *e-mail: farizimranov@yandex.ru **e-mail: kobelkov@dodo.inm.ras.ru ***e-mail: shurunya@mtu-net.ru