  Alfv\'enic fluctuations in the solar wind display many properties reflecting
an ongoing nonlinear cascade, e.g. a well-defined spectrum in frequency,
together with some characteristics more commonly associated with the linear
propagation of waves from the Sun, such as the variation of fluctuation
amplitude with distance, dominated by solar wind expansion effects. Therefore
both nonlinearities and expansion must be included simultaneously in any
successful model of solar wind turbulence evolution. Because of the disparate
spatial scales involved, direct numerical simulations of turbulence in the
solar wind represent an arduous task, especially if one wants to go beyond the
incompressible approximation. Indeed, most simulations neglect solar wind
expansion effects entirely. Here we develop a numerical model to simulate
turbulent fluctuations from the outer corona to 1 AU and beyond, including the
sub-Alfv\'enic corona. The accelerating expanding box (AEB) extends the
validity of previous expanding box models by taking into account both the
acceleration of the solar wind and the inhomogeneity of background density and
magnetic field. Our method incorporates a background accelerating wind within a
magnetic field that naturally follows the Parker spiral evolution using a
two-scale analysis in which the macroscopic spatial effect coupling
fluctuations with background gradients becomes a time-dependent coupling term
in a homogeneous box. In this paper we describe the AEB model in detail and
discuss its main properties, illustrating its validity by studying Alfv\'en
wave propagation across the Alfv\'en critical point.
