  Collective flow observed in heavy ion collisions is largely attributed to
initial geometrical fluctuations, and it is the hydrodynamic evolution of the
system that transforms those initial spatial irregularities into final state
momentum anisotropies. Cumulant analysis provides a mathematical tool to
decompose those initial fluctuations in terms of radial and azimuthal
components. It is usually thought that a specified order of azimuthal cumulant,
for the most part, linearly produces flow harmonic of the same order. In this
work, we carry out a systematic study on the connection between cumulants and
flow harmonics using a hydrodynamic code called NeXSPheRIO. We conduct three
types of calculations, by explicitly decomposing the initial conditions into
components corresponding to a given eccentricity and studying the out-coming
flow through hydrodynamic evolution. It is found that for initial conditions
deviating significantly from Gaussian, such as those from NeXuS, the linearity
between eccentricities and flow harmonics partially breaks down. Combining with
the effect of coupling between cumulants of different orders, it causes the
production of extra flow harmonics of higher orders. We argue that these
results can be seen as a natural consequence of the non-linear nature of
hydrodynamics, and they can be understood intuitively in terms of the
peripheral-tube model.
