An algorithm for calculating Hermite-based finite difference weights

Finite difference (FD) formulas approximate derivatives by weighted sums of function values. Given arbitrarily distributed node locations in one-dimension, a previous algorithm by the present author (1988, Generation of finite difference formulas on arbitrarily spaced grids. Math. Comput., 51, 699-706) provides FD weights of optimal order of accuracy for approximating any order derivative at a specified location. This algorithm is extended here to the case of finding weights to apply not only to function values but also to first derivative values in the case that these also are available. The MATLAB code for the algorithm is provided, and two examples are given to illustrate how this type of FD stencil can be applied to solving partial differential equations.

Automated summary

The algorithm provides FD weights of optimal accuracy for approximating any order derivative at a specified location with arbitrarily distributed node locations in one-dimension, and can now also be applied to first derivative values. The MATLAB code for the algorithm is provided, with two examples illustrating its application in solving partial differential equations.