# ENO/WENO vs monotone Hermite interpolation

I have see the method PCHIP in matlab that implements the monotone Hermite interpolation method which was originally proposed by Carlson in 1980s. It seem to accomplish the goal of preventing the values go outside of the range. I have not seen the error estimates results but I guess it does as good as cubic polynomial and locally $O(h^4)$. Now, there are more recent ENO/WENO methods and their multiple "kids". Would like to hear why these methods stand out and why they are "better" or worse compare to monotone Hermite?

• In ENO (Essentially Non Oscillatory) methods, the smoothest stencil from all the candidate stencils is chosen (based on Newton's Divided Differences). In WENO (Weighted ENO) schemes, all the candidate stencils are used with appropriate weights attached to them (zero for discontinuity and optimum for smooth portion). ENO and WENO schemes are useful in the cases where solution contains discontinuities. – Subodh Jan 11 '13 at 6:01

• but if the solution of $u_t+cu_x=0, x\in [0,2]$ is not very discontinuous $u_0=1_{\{x\leq 1\}}$, and has a jump at one point only, if make sure that the grid is constructed in such a way that the discontinuity falls at the grid point at each time level, in that case does PCHIP work as well? – Kamil Jan 12 '13 at 14:42