A piecewise-quintic interpolation scheme (Q1815957)

A piecewise-quintic interpolation scheme, previously defined for the advection problem, is substantially developed, and a complex shape-preserving analysis, as well as new numerical results, are presented. The scheme considered is based on a four-point grid stencil, defined on a uniform mesh. The interpolant utilizes four consecutive grid data points and the first derivative estimates at the internal points. Sufficient conditions for the scheme to be positive definite are formulated in terms of the discrete maximum principle. Monotonicity conditions are characterized as admissible variability regions of the respective scheme's parameters. Standard limiter functions for derivative estimates are applied with accuracy gain obtained by relaxing monotonicity constraints near local extremes. Results of numerical tests are presented for regular function interpolation as well as for 1D and 2D advection of standard test profiles.