COMPLETELY CONSERVATIVE DIFFERENCE

tion- of mass, momentum and energy. In the numerical calculation of gas dynamical problems by the method of finite differences the system of differential equations is approximated by a difference scheme, which is equivalent to the replacement of the continuous medium by some discrete model of it. This model must reflect the fundamental properties of the medium. Therefore, it is natural in the first place to require that the corresponding difference analogs of the conservation laws be satisfied, that is, that the difference scheme be conservative. A. N. Tikhonov and A. A. Samarskii directed attention to this important fact at the beginning of the fifties. An example was constructed [ll, where a non-conservative difference scheme, having a second order of approximation to the smooth solutions, diverges in the case of a discontinuous solution of the differential equation. It is usually considered that to obtain a conservative difference scheme it is sufficient to approximate the three fundamental laws of conservation (balance) (see [2, 31, chapter III of the book [41, where a review is given of references on the numerical methods of gas dynamics). However, there is a theoretical aspect here. In the system of equations of gas dynamics the energy equation can be written in one of two forms- divergent and nondivergent. In differential form these forms are completely equivalent and may be transformed into one another by means of the remaining equations of the differential system. Therefore, for the system of equations of gas dynamics in differential form both the law of conservation of the total energy, and the balance of the internal energy are simultaneously valid and imply one another