The quantitative description of the confined quantum scattering in waveguide-like atomic traps is an actual problem of modern physics of ultracold atoms and molecules. It is a challenging problem of computational mathematics to integrate the few-dimensional Schrödinger equation describing such scattering. In the present work we show how the split-operator method developed by V.S. Melezhik in discrete-variable representation can be parallelized and extended for calculation of probability density distribution in such quantum systems. By using as an example the confined collision of Li atoms with Yb ions in a hybrid atom-ion trap we demonstrate calculation of the time-evolution of the atom-ion probability density distribution. The calculated function is an important parameter for analysis of this reaction. Due to resent development of unique experimental technique in this field it becomes actual experimental analysis of cold low-dimensional few-body systems. However, interpretation and planning of the experiments demand quantitative description of the systems. The present work opens promising perspective in the development of this direction. © 2019, Springer Nature Switzerland AG.