Conservative difference schemes of nonstationary gas dynamics

As a rule, equations of gas dynamics are discontinuous. From a physical point of view it is fairly common to distinguish weak discontinuities relating to cutting waves and strong discontinuities relating to shock waves . For these reasons successive grid refinement can be made with caution when the accurate account of accuracy of numerical methods is performed. 

In this section we initiate the design of difference methods for numerical solutions of the simplest problems in gas dynamics. Of our initial concern is the problem about one-dimensional non,stationary gas flow in a plane with the following ingredients velocity v, density p, temperature T, pressure p, internal energy e. 

In preparation for this, the equations of gas dynamics will reproduce the conservation laws of impuls, mass and energy that can be written in a number of different ways with respect to Eulerian (x,t) or Lagrangian (s,f) variables, where x is the coordinate of a particle and s is the initial coordinate of a particle or the quantity 

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A combination of the second and third equations we have mentioned above gives 

In this context, two limiting cases of interest are as follows  

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