N. Mohankumar and Scott M. Auerbach,
``On the Use of Higher-Order Formulas for Numerical Derivatives
in Scientific Computing,'' Comput. Phys. Commun. 161,
109-118 (2004).
Abstract
In many situations, the numerical derivative of a function at a point
x must be calculated since the function is not defined by a closed-form
expression, but rather by values of the function at grid points at and
around x. This typically arises when enforcing the boundary conditions
while solving a differential equation. Usually, one employs a 2- or
3-point formula to approximate the derivative. On the other hand,
the use of a higher-order formula, such as a 7- or even a 10-point
approximation, based on the method of undetermined coefficients,
can sometimes lead to better accuracy and enhanced computational
efficiency. We show that significant improvements arise from using
higher-order formulas for the first derivative in two important
problems: the calculation of quantum mechanical reaction rates using
the Miller-Schwartz-Tromp correlation function, and the calculation
of the radioactivity migration in a porous medium.
Prof SM Auerbach
22 July 2004