In this paper we numerically solve the stationary inviscid flow around an airfoil. Using the second-order explicit Runge¿Kutta method in combination with the MUSCL scheme and the minmod limiter, we do not obtain a machine accurate solution. This has already been observed in literature and is explained by the non-differentiability of the minmod limiter. An analysis of the limiter shows that it is possible to obtain a machine accurate solution with an asymmetric minmod limiter if an implicit scheme with low CFL number is used. For higher CFL number the convergence rate of this scheme increases considerably at the expense of a strong increase in the final residual level. A further study of the differences reveals that the steady state obtained with the implicit method is in fact unstable and can only be found due to the dissipation present in the implicit method. In this paper we give some evidence that the stall in convergence with the explicit method might be caused by a physical instability in the wake behind the airfoil. This instability is also predicted by linear stability theory and confirmed by a grid refinement study.
van Buuren, R., Kuerten, J. G. M., & Geurts, B. J. (1997). Instabilities of stationary inviscid flow around an airfoil. Journal of computational physics, 138(2), -. https://doi.org/10.1006/jcph.1997.5830