Shape-preserving subdivision scheme with the third-order accuracy and C² smoothness

In this study, we present a novel shape preserving C² subdivision scheme with third-order accuracy. Its limit functions preserve both monotonicity and convexity of the given data, even in cases where the data are non-strictly monotone or convex. To achieve this, we especially devise a modified minmod method, originally introduced in Gelb and Tadmor (2006) to detect edges from a piecewise smooth data, that plays a role of limiting procedure to prevent spurious oscillations. While most of shape preserving schemes are complicated, the proposed method is conceptually simple to implement. Some numerical results are presented to demonstrate the accuracy, smoothness and shape preserving performance of the proposed scheme.