  Several finite difference methods are proposed for the infinitesimal
generator of 1D asymmetric $\alpha$-stable L\'{e}vy motions, based on the fact
that the operator becomes a multiplier in the spectral space. These methods
take the general form of a discrete convolution, and the coefficients (or the
weights) in the convolution are chosen to approximate the exact multiplier
after appropriate transform. The accuracy and the associated
advantages/disadvantages are also discussed, providing some guidance on the
choice of the right scheme for practical problems, like in the calculation of
mean exit time for random processes governed by general asymmetric
$\alpha$-stable motions.
