  Non-Gaussian L\'evy noises are present in many models for understanding
underlining principles of physics, finance, biology and more. In this work, we
consider the Fokker-Planck equation(FPE) due to one-dimensional asymmetric
L\'evy motion, which is a nonlocal partial differential equation. We present an
accurate numerical quadrature for the singular integrals in the nonlocal FPE
and develop a fast summation method to reduce the order of the complexity from
$O(J^2)$ to $O(J\log J)$ in one time-step, where $J$ is the number of unknowns.
We also provide conditions under which the numerical schemes satisfy maximum
principle. Our numerical method is validated by comparing with exact solutions
for special cases. We also discuss the properties of the probability density
functions and the effects of various factors on the solutions, including the
stability index, the skewness parameter, the drift term, the Gaussian and
non-Gaussian noises and the domain size.
