Conservative difference schemes for one-dimensional flows of polytropic gas

The paper considers one-dimensional flows of polytropic (calorically ideal) gas. These flows include three cases of gas dynamics: plain one-dimensional flows (one-dimensional space), radially symmetric flows in two-dimensional space and spherically symmetric flows in three-dimensional space. Starting with the difference schemes which have conservation laws of mass and energy (as well as conservation of momentum and the center of mass motion for the plain one-dimensional flows), we find difference schemes which also have additional conservation laws for the special values of the adiabatic exponent $\gamma = 1 + 1 /d $, where $d$ is the space dimension.