  In this paper we discuss the adiabatic projection method, a general framework
for scattering and reaction calculations on the lattice. We also introduce
several new techniques developed to study nucleus-nucleus scattering and
reactions on the lattice. We present technical details of the method for
large-scale problems. To estimate the systematic errors of the calculations we
consider simple two-particle scattering on the lattice. Then we benchmark the
accuracy and efficiency of the numerical methods by applying these to calculate
fermion-dimer scattering in lattice effective field theory with and without a
long-range Coulomb potential. The fermion-dimer calculations correspond to
neutron-deuteron and proton-deuteron scattering in the spin-quartet channel at
leading order in the pionless effective field theory.
