@TechReport{ it:2012-018,
author = {Bengt Fornberg and Erik Lehto and Collin Powell},
title = {Stable Calculation of {G}aussian-based {RBF-FD} Stencils},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2012},
number = {2012-018},
month = aug,
abstract = {Traditional finite difference (FD) methods are designed to
be exact for low degree polynomials. They can be highly
effective on Cartesian-type grids, but may fail for
unstructured node layouts. Radial basis function-generated
finite difference (RBF-FD) methods overcome this problem
and, as a result, provide a much improved geometric
flexibility. The calculation of RBF-FD weights involves a
shape parameter $\varepsilon$. Small values of
$\varepsilon$\ (corresponding to near-flat RBFs) often lead
to particularly accurate RBF-FD formulas. However, the most
straightforward way to calculate the weights (RBF-Direct)
then becomes numerically highly ill-conditioned. In
contrast, the present algorithm remains numerically stable
all the way into the $\varepsilon\rightarrow0$ limit. Like
the RBF-QR algorithm, it uses the idea of finding a
numerically well-conditioned basis function set in the same
function space as is spanned by the ill-conditioned
near-flat original Gaussian RBFs. By exploiting some
properties of the incomplete gamma function, it transpires
that the change of basis can be achieved without dealing
with any infinite expansions. Its strengths and weaknesses
compared the Contour-Pad{\'e}, RBF-RA,\ and RBF-QR
algorithms are discussed.}
}