  We report on a new Monte Carlo method for simulating diffusive shock
acceleration (DSA) of solar energetic particles at upstream and downstream
regions of quasi-parallel collisionless shock waves under the influence of
self-generated turbulence. By way of example, we apply the model to a fast 1500
km \mathrm{s}^{-1} coronal mass ejection at ten solar radii. Results indicate
that the maximum energies at outer corona are likely to be limited to few MeV,
due to lack of suprathermal protons for appreciable wave growth, and
insufficient time required acceleration. We find that the second-order Fermi
acceleration, although being a too slow process to have a notable effect at the
highest energies, significantly flattens energy spectra at low energy end.
Simulations indicate that protons continue to damp waves efficiently several
solar radii from the shock in the downstream region, which may be an important
mechanism for heating suprathermals. Our simulations also suggest that models
assuming a simple isotropic scattering are likely to predict too efficient
acceleration.
