The time evolution of the ground state of the hydrogen atom in a very intense, linearly polarized, pulsed laser field is investigated by solving the two-dimensional Schrodinger equation. We use a unitary propagation scheme based on multiple application of the split operator formalism. For a realistic description, the Coulomb singularity is not cut off but treated exactly using fast sine transforms to switch between direct and Fourier space. The implementation and performance enhancement through parallelization on a Gray T3E are discussed. (C) 1999 Elsevier Science B.V. All rights reserved.