Euler equations of generalized Riemann variable are derived from unsteady primitive variable Euler equations and solved by using two - a point-two-step upwind finite difference method.

The hyperbolic equations were formulated by artificial compressibility method with the convective terms discreted using a third-order upwind scheme based on Roe's approximate Riemann solver.

To solve it numerically, we use a moving mesh method, two upwind difference discretizations (simple upwind scheme and midpoint upwind scheme) are applied.