Direct Numerical Solution of the Steady 1D Compressible Euler Equations for Transonic Flow Profiles with Shocks

It is well-known that stationary transonic solutions of the compressible Euler equations are hard to compute using the stationary form of the equations. Therefore, time marching methods with explicit or implicit time integration are normally employed. In this paper a method is described that computes one-dimensional transonic flows directly from the stationary equations. The method is based on a dynamical systems formulation of the problem and uses adaptive integration combined with a 2×2 Newton method. Example calculations show that the resulting method is fast and accurate. A sample application area for this method is the calculation of transonic hydrodynamic escape flows from extrasolar planets and the early Earth, and the method is also illustrated for quasi-one-dimensional flow in a converging-diverging nozzle. The method can be extended easily to handle flows with shocks, using a Newton method applied to the Rankine-Hugoniot relations in order to match the shock location with the outflow boundary condition.