An Accurate P$^{3}$M Algorithm for Gravitational Lensing Studies in
  Simulations
  We present a two-dimensional (2D) Particle-Particle-Particle-Mesh (P$^3$M)
algorithm with an optimized Green function and adaptive softening length for
gravitational lensing studies in N-Body simulations. The analytical form of the
optimized Green function $\hat{G}(\rm{k})$ is given. The softening schemes
($S$) are studied for both the PM and the PP calculations in order for accurate
force calculation and suppression of the particle discreteness effect. Our
method is two orders of magnitude more accurate than the simple PM algorithm
with the {\it poor man's} Green function ($\propto1/k^2$) at a scale of a few
mesh cells or smaller. The force anisotropy is also much smaller than the
conventional PM calculation. We can achieve a force accuracy better than 0.1
percent at all scales with our algorithm, which makes it an ideal (accurate and
fast) algorithm for {\textit{micro}} lensing studies . When we apply the
algorithm to computing {\textit{weak}} and {\textit{strong}} lensing quantities
in N-Body simulations, the errors are dominated by the Poisson noise caused by
particle discreteness. The Poisson noise can be suppressed by smoothing out the
particle distribution, which can be achieved by simply choosing an adaptive
softening length in the PP calculation. We have presented a criterion to set
the adaptive softening length. Our algorithm is also applicable to cosmological
simulations. We provide a \textsc{python} implementation \texttt{P3Mlens} for
this algorithm.
