The major problem related to those monotonicity preserving techniques is a loss of accuracy in the vicinity of
In the paper a piecewise quintic polynomial interpolation scheme, based on a four-point stencil and a uniform grid is investigated. strict local extrema. The ''clipping'' effect observed results
The interpolant utilizes four consecutive grid data points and the from the property that the algorithm still produces a monofirst derivative estimates at the internal points. Sufficient conditions tone interpolant, even if the data are no longer monotone.
for the scheme to be positive definite are formulated in terms of
In [10] a class of accurate, piecewise cubic schemes is conthe discrete maximum principle. Monotonicity conditions are characterized as admissible variability regions of the respective sidered, where the gain of the overall precision is obtained scheme's parameters. Standard limiter functions for derivative estiby replacing standard monotonicity conditions by high ormates are applied with accuracy gain obtained by relaxing monotonder derivative estimates in the neighborhood of local exicity constraints near local extrema. Results of numerical tests are trema. The additional mesh points must be utilized in that presented for regular function interpolation as well as for 1D and 2D advection of standard test profiles.