Approximations to Function Values and First Derivatives from Sets of Integral-Averaged Function Values

摘要

Stimulated by the necessity of numerically integrating the one-dimensional, first-order, magnetic-flux-surface-averaged particle continuity equation to study impurity transport in tokamaks, we have investigated how to obtain estimates of functions and their first derivatives correct to second order in the grid spacing for arbitrary grids. Using integral-averaged values with arbitrary weight functions, we readily find two-term linear approximations to the functions that satisfy the accuracy criterion. However, in the case of two-term linear approximations to the first derivative, we find that the grid partition must satisfy a nodal equation to obtain the desired accuracy. When the entire grid cannot be set up to satisfy the nodal equation, transitions involving three-term linear approximations to the derivative are required. We also determine the point between two grid points at which the integral-averaged function value can be identified with the function evaluated at that point. Finally, we give examples of grid selection for two different weight functions: w = 1 and w = r. (ERA citation 08:003733)