Roe (1981):

- Scheme (essentially) as above.
- With clever choice of matrix , so that the approximate Riemann solver becomes exact for simple waves (only one shock or only one contact discontinuity).
- Needs entropy fix.
- The fluctuations are taken from the primitive rather than the conservative variables.

HLLE (1983, 1988):

- Uses only the two waves with largest and smallest characteristic velocity.
- The region in between is approximated by a single state.
- The scheme is simple, more robust but also more dissipative than Roe's scheme.