Shape Preserving Third and Fifth Degrees Polynomial Splines

Abstract

This paper is devoted to the development of positivity and monotonisity preserving linear spline techniques, namely, techniques which are based on ideas applied in the field of high order TVD (Total Variation Diminishing) methods for numerical solving compressible flow equations. Third and fifth degrees polynomial splines are constructed. Third degree splines include two variants, namely, monotonisity preserving and positivity preserving splines. These splines may be considered as modifications of classical cubic spline and may be identical to this spline for "good" data. These splines get shape preserviation at the cost of reducing smoothness till C^1. To restore C^2smoothness fifth degree polynomial splines are considered, which are constructed as a sum of base cubic shape preserving splines and fifth degree terms, which are chosen to provide continuity of the spline second derivative. These C^2fifth degree polynomial splines are observed to preserve monotonisity or positivity for all considered data with these properties.

Weighted Fifth Degree Polynomial Spline

“…Monotonisity preserving spline is constructed in [13,14] by delimiting some terms in equations (4). Next function-delimitator is used:…”

“…To increase the spline smoothness and to provide continuity of the spline second derivative, a technique written in [14] is used here, namely, the fifth degree term is added to the cubic polynomial. Let next formulae is used instead of the formulae (1):…”

“…Ideas of algorithms of the first family are applied to constructing of monotonisity or positivity preserving polynomial splines in [13,14]. Paper [15] is devoted to the application of algorithms of the second family to constructing of global polynomial splines, reducing undesirable oscillations.…”