Two-step shock waves propagation for isothermal Euler equations

The feedback control algorithm is applied to provide stable propagation of a two-step shock waves for nonlinear isothermal Euler equations despite the desired profile and velocity of the waves do not correspond to an analytical solution of the equations. Two cases are considered: transition to the two-step shock wave solution form the usual one-step wave and generation of a wave with a two-step front from an initially undisturbed velocity field. In both cases arising of two-step shock waves is obtained and an influence of the control algorithm coefficients on the shape of the waves is established.