At some point only a finer grid helps:
- (W)ENO schemes can have very high order.
- (W)ENO schemes can be complemented by high-order Runge-Kutta steps.
- PPM: non-linear criteria can help to decide if e.g. shocks should be sharpened further.
- Subcell resolution can resolve the position of discontinuities within cells.
- More grid points
- Non-equidistant grid
- Adaptively refined grid