A particular case of initial data for the two-dimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well-posed.
Elling, V. (2006). A possible counterexample to well posedness of entropy solutions and to Godunov scheme convergence. Mathematics of Computation, 75(256), 1721–1733. https://doi.org/10.1090/s0025-5718-06-01863-1