High-performance GPU parallel solver for 3D modeling of electron transfer during ion-surface interaction

This study presents parallelized time-dependent Schrödinger equation solver (GPU TDSE Solver) for 3D modeling of resonant electron transfer (RCT) during ion-surface interactions and atomic collisions. The computer modeling of RCT process is based on the numerical one-electron TDSE solution in relatively large spatial domain (about 103-104 nm3). Due to the numerical complexity of direct 3D TDSE solution in such domains, most of RCT calculations use approximations that reduce problem to 2D calculations (e.g. cylindrical symmetry). Last years the TDSE Solver was developed for 3D RCT modeling in large-scale nanosystems (Gainullin and Sonkin, 2015). It was shown to have rather good performance due to the effective parallel implementation of simple numerical scheme on GPUs (explicit finite-difference method in Cartesian coordinates). Note, that usage of FDM in Cartesian coordinates requires ∼100 times greater numerical grid, comparing to the finite-element or finite-volume methods. For the majority of RCT problems the calculations transfer to the cylindrical coordinates decreases the size of numerical grid by an order of magnitude without loss of the calculations precision. The main problem in the transfer to the cylindrical coordinates is that explicit numerical schemes for parabolic equations, including TDSE, are unstable near the axis ρ=0. This study presents hybrid numerical scheme, which eliminates this instability and preserves the effective parallelization on GPUs. The performance of new version of the GPU TDSE Solver, based on the hybrid numerical scheme, was found to be 6 times greater comparing to the previous version. Such performance gain is stipulated by less discrete points, required for the FDM implementation in cylindrical coordinates. Due to reduction of required GPU memory, new version of GPU TDSE Solver can handle spatial domains about 103 nm3 using an ordinary personal computer (equipped with modern GPU, e.g. Tesla k20 or better) or up to 105 nm3 using supercomputers. GPU TDSE Solver was applied to the calculation of the resonant charge transfer in nanosystems. The calculated neutralization probability for Li+ ions impinging on the Ag(100) surface shows a good quantitative agreement with the experimental data.