Relaxation methods for steady transonic flow calculations by Euler equations

Transonic potential equations are mixed elliptical-hyperbolic, but Euler equations are mixed hyperbolic. Classic relaxation methods cannot be used for the hyperbolic equation. The natural techniques for solving the discretized equations corresponding to hyperbolic equations is marching in one of the hyperbolic directions. Since Euler equations are hyperbolic with respect to time, the obvious marching direction is the time direction. Time marching schemes have at best a convergence rate that is one order of magnitude lower than relaxation methods. An extension of relaxation schemes to hyperbolic equations allows the construction of algorithms for Euler equations which have the same convergence rates as the relaxation schemes for the potential equation.