On a certain approach to constructing difference schemes for quasilinear equations of gas dynamics (Q1975807)

The authors consider a time-dependent initial-boundary value problem for a system of gas dynamics equations which describes the three-dimensional motion of inviscid non-heat-conducting gas. As is mentioned in \textit{A. Harten} [J. Comput. Phys. 49, 151-164 (1983; Zbl 0503.76088)], such a system can be written in a symmetric \(t\)-hyperbolic form: \[ B^0(Q)\cdot Q_t+ \sum^3_{k=1} B^k(Q)\cdot Q_{x_k}= 0,\tag{1} \] where \(B^0\), \(B^k\) are symmetric matrices of order 5, \(B^0> 0\), and \(Q= (q_1,\dots, q_5)^*\) is the vector of dependent variables. In the book by \textit{A. M. Blokhin} and \textit{R. D. Alaev} [Energy integrals and their applications to the study of stability of difference schemes (Russian), Novosibirsk, Novosibirsk State University (1993)] it is shown that, with a special choice of the vector \(Q\), system (1) can be rewritten in the following equivalent form: \[ (B^0(Q)\cdot Q)_t+ \sum^3_{k= 1} (B^k(Q)\cdot Q)_{x_k}= 0.\tag{2} \] These two variants of the original initial system, equivalent on smooth solutions, allow the authors to derive a priori local estimates of the solution. In the article under review, a class of implicit difference schemes is described whose construction is based on the fact that the system of gas dynamics admits two versions, (1) and (2). This fact is used to obtain energy estimates for approximate solutions which imply stability of schemes considered. As an example, the authors present a scheme for one-dimensional model describing charge transport in semiconductors. Results of calculations are not presented.