  To increase the reliability of simulations by particle methods for
incompressible viscous flow problems, convergence studies and improvements of
accuracy are considered for a fully explicit particle method for incompressible
Navier--Stokes equations. The explicit particle method is based on a penalty
problem, which converges theoretically to the incompressible Navier--Stokes
equations, and is discretized in space by generalized approximate operators
defined as a wider class of approximate operators than those of the smoothed
particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods.
By considering an analytical derivation of the explicit particle method and
truncation error estimates of the generalized approximate operators, sufficient
conditions of convergence are conjectured.Under these conditions, the
convergence of the explicit particle method is confirmed by numerically
comparing errors between exact and approximate solutions. Moreover, by focusing
on the truncation errors of the generalized approximate operators, an optimal
weight function is derived by reducing the truncation errors over general
particle distributions. The effectiveness of the generalized approximate
operators with the optimal weight functions is confirmed using numerical
results of truncation errors and driven cavity flow. As an application for flow
problems with free surface effects, the explicit particle method is applied to
a dam break flow.
