COMPLETELY CONSERVATIVE DIFFERENCE SCHEMES FOR THE EQUATIONS OF GAS DYNAMICS IN EULER’S VARIABLES*

1. THE set of equations of gas dynamics can be written in a variety of ways, each capable of direct physical interpretation. For instance, the energy equation may be written in the divergent form, expressing the variation of the total energy. or in the non-divergent form, expressing the variation of the internal energy only, or in the entropic form etc. These forms are equivalent in the differential form, in the sense that one can be obtained from the other by means of the remaining equations of the set. When the differential equations are solved numerically by the method of finite differences, it is approximated by a difference scheme. The latter may be based on any one of the equivalent forms of the differential equations. In particular, classical conserva-tive schemes are based on the divergent equations [ 11. But the equivalence property does not in general hold for the set of difference equations. For instance, if the non-divergent energy equation is used in the scheme, it cannot always be reduced to the divergent difference form. In the course of transformation, due to “mismatch ” of the individual equations of the scheme, remainder terms appear, the presence of which may be treated as due to energy sources of a purely difference type being present in the scheme. These fictitious sources lead to violation of the difference analogue of the law of conservation of total energy. Similarly, the divergent difference equation cannot be transformed into the corre-sponding non-divergent form. In this difference scheme, when the law of conservation of total energy is satisfied, the internal energy, and hence the kinetic energy, balance is destroyed. The energy unbalance depends on the nature of the solution. While it is small on smooth functions, it can become comparable to the total energy on solutions which vary strongly in time and space