  Aims. We aim to improve a previous model for the prediction of column
densities and deuterium fractions of non- and singly deuterated methanol.
Thereby, we try to identify crucial chemical and physical parameters, for which
the study of deuteration could provide valuable additional constraints.
  Methods. We employed a gas-grain chemical code to devise a model that is in
agreement with the observed column density and deuterium fraction profiles of
the innermost region of the pre-stellar core L1544. For that purpose, we
developed a new treatment of reactive desorption, deriving an individual
reactive desorption efficiency for every product species in a chemical
reaction, that depends on the reaction enthalpy and type of underlying surface.
Furthermore, we explored several options to promote the diffusion of hydrogen
and deuterium atoms over the surface of interstellar dust grains, in order to
increase methanol formation.
  Results. Our fiducial model employs diffusion by quantum tunneling of
hydrogen and deuterium atoms, resulting in CH$_3$OH and CH$_2$DOH column
densities that are approximately an order of magnitude lower than the observed
values, which improves the results compared to the previous model by a factor
10. The $N$(CH$_2$DOH)/$N$(CH$_3$OH) ratio is reproduced within a factor of 1.2
for the centre and 1.8 for the position of the methanol peak. Given the large
uncertainties that chemical models typically have, we consider our predictions
to be in agreement with the observations. In general, we conclude that a
diffusion process with a high diffusion rate needs to be employed to obtain
methanol column densities that are in accordance with the observed values.
Also, we find that the introduction of abstraction reactions into the methanol
formation scheme suppresses deuteration, when used in combination with a high
diffusion rate.
