Shape-Preserving Local Interpolation for Plotting Solutions of ODEs

We investigate the use of piecewise rational interpolants of Delbourgo and Gregory in an important and widely occurring application. We propose the following algorithm for visually pleasing plots of the solution of an ordinary differential equation (ODE): use piecewise cubic Hermite interpolation where it can be shown to preserve shape (monotonicity and/or convexity) and also where there is no shape to preserve, otherwise use the appropriate convex or monotone piecewise rational interpolant. Bounds are derived which enable efficient plotting of the rational interpolants. This scheme should be useful in any context where both solution and derivative of a function are available as data.