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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?

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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 at 6:01

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PCHIP is not a conservative reconstruction, making it inappropriate for conservation laws. Furthermore, hyperbolic problems have discontinuous solutions so there is generally no benefit to a continuous reconstruction. Conservative monotone spline reconstructions are being investigated by the UK Met Office for use in tracer advection for atmosphere modeling, see papers on the multi-dimensional case, quartic splines, and applications. These methods are relatively new and are not currently popular with many other groups. Some reasons for this include

  • nonlocal reconstruction is inconvenient, especially in parallel
  • these methods are semi-Lagrangian and currently only suitable for advection, especially in multiple dimensions
  • there is no characteristic spline-based reconstruction
  • only structured grids can be used
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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 at 14:42

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