  Open Computing Language (OpenCL) is a parallel processing language that is
ideally suited for running parallel algorithms on Graphical Processing Units
(GPUs). In the present work we report on the development of a generic parallel
single-GPU code for the numerical solution of a system of first-order ordinary
differential equations (ODEs) based on the OpenCL model. We have applied the
code in the case of the Time-Dependent Schr\"odinger Equation of atomic
hydrogen in a strong laser field and studied its performance on NVIDIA and AMD
GPUs against the serial performance on a CPU. We found excellent scalability
and a significant speed-up of the GPU over the CPU device which tended towards
a value of about 40 with significant speedups expected against multi-core CPUs.
