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.