  The Galileon scalar field theory is a prototypical example of an effective
field theory that exhibits the Vainshtein screening mechanism, which is
incorporated into many extensions to Einstein gravity. The Galileon describes
the helicity zero mode of gravitational radiation, the presence of which has
significant implications for predictions of gravitational waves from orbiting
objects, and for tests of gravity sensitive to additional polarizations.
Because of the derivative nature of their interactions, Galileons are
superficially not well-posed as effective field theories. Although this
property is properly understood merely as an artifact of the effective field
theory truncation, and is not theoretically worrisome, at the practical level
it nevertheless renders numerical simulation highly problematic.
Notwithstanding, previous numerical approaches have successfully evolved the
system for reasonable initial data by slowly turning on the interactions. We
present here two alternative approaches to improving numerical stability in
Galileon numerical simulations. One of these is a minor modification of
previous approaches, which introduces a low pass filter that amounts to
imposing a UV cutoff together with a relaxation method of turning on
interactions. The second approach amounts to constructing a (numerical) UV
completion for which the dynamics of the high momentum modes is under control,
and for which it is unnecessary to slowly turn on nonlinear interactions. We
show that numerical simulations of the UV theory successfully reproduce the
correct Galileon dynamics at low energies, consistent with the low-pass filter
method and with previous numerical simulations.
