Second order accurate upwind solutions of the two-dimensional steady Euler equations by the use of a defect correction method

Finite volume discretization of the steady Euler equations by the finite volume technique is discussed. On an irregular mesh it is shown how to apply Van Leers projection-evolution stages in the discretization. The rotational invariance of the Euler equations is used. For a general numerical flux function, consistent with the physical flux, a proof of the order of accuracy for a first and second order upwind scheme is given. Results hold for all well known approximate Riemann-solvers. Second order accurate approximations are obtained by a defect correction (DeC) method. A limiter, used in the DeC method, is constructed to maintain monotone solutions. For two typical model problems (an oblique shock and a contact discontinuity), only a few (3 or 4) DeC iteration steps are sufficient to steepen discontinuities effectively. This makes the method cheap to apply. The quality of the results seems comparable with results obtained by TVD schemes.