Calculation of the Vacuum Green's Function Valid even for High Toroidal Mode Number in Tokamaks
Authors: M.S. Chance, A.D. Turnbull, and P.B. Snyder
Date of PPPL Report: February 2006
Submitted to: Journal of Computational Physics
The present evaluation of the Green's function used for the magnetic scalar potential in vacuum calculations for axisymmetric geometry in the vacuum segments of GATO, PEST and other MHD stability codes has been found to be deficient even for moderately high toroidal mode numbers, n, relevant to the edge localized peeling-ballooning modes. This was due to the loss of numerical precision arising from the upward recursion relation used for generating the functions from the values at n = 0 from the complete elliptic integrals of the first and second kinds. To ameliorate this, a direct integration of the integral representation of the function is crafted to achieve the necessary high accuracy for moderately high mode numbers, with due consideration to the singular behavior of the integrand involved. At higher mode numbers the loss of numerical precision due to cancellations from the oscillatory behavior of the integrand is further avoided by judiciously deforming the integration contour into the complex plane to obtain a new integral representation for the Green's function. Near machine precision, roughly 12–16 digits, can be achieved by using a combination of these techniques. The relation to the associated Legendre functions is also described.