Monotone \(C^2\) Quartic Spline Method with Non-Oscillation Derivatives: A Fundamental Approach - Yao, Jin; Nelson, Karl E. (Mathematics and Computer Science: Contemporary Developments Vol. 3, 2024)
A one-dimensional monotone interpolation based on interface reconstruction with partial volumes in the slope-space utilizing the Hermite cubic-spline is proposed. The new method is only quartic, however is C2 and unconditionally monotone. A set of control points in addition to the data points is employed to constrain the curvature of the interpolation function and to eliminate possible nonphysical oscillations in the slope space. The slope f(x)=g'(x) of an interpolation polynomial g(x) is treated as the area density over the domain, such that g(x) represents the area under the curve y = f(x) in the derivative space. At each data point xi, f'(xi) can be estimated by differentiating a parabola fit using three data points including left and right neighbors. Within each interval between a pair of neighbor data points, a second parabola fit constrained by the ‘area’ and the slopes at the endpoints, is used to compute the value of f(xi). To ensure continuity of f'(x) at each xi, an additional control point is introduced in the middle of each interval. A set of Hermit spline, which matches the area in each interval and passes each point (xi,f(xi)), is then utilized. The positions of these control points are determined by solving a tri-diagonal linear system, which is considered explicit. The solution of monotone interpolation g(x) is finally obtained by integrating the Hermit spline.