دانلود رایگان مقاله حل کننده GPU برای مدل سازی سه بعدی انتقال الکترون در طول برهمکنش سطح یونی

Abstract

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.

5. Conclusion

The new version of GPU TDSE Solver for 3D modeling of resonant electron transfer (RCT) during ion–surface interactions and atomic collisions is presented in this study. Note, that the key requirement to the TDSE Solver’s for 3D RCT modeling is capability to handle large spatial domains (about 103 –104 nm3 ). Contrary to the previous version, the new version of GPU TDSE Solver performs the FMD explicit scheme in cylindrical coordinates, instead of Cartesian. The calculations transfer to the cylindrical coordinates significantly decreases the size of numerical grid. The main problem in the transfer to the cylindrical coordinates is that explicit numerical schemes for parabolic equations, including TDSE, are unstable near the axe ρ = 0. In the present study, the hybrid numerical scheme for parabolic PDEs (including TDSE) in cylindrical coordinates was proposed, which eliminates instability near axe ρ = 0 and preserves effective parallelization of the explicit leap-frog numerical scheme. 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 and 20 times greater comparing to other known TDSE solvers. Such performance gain is obtained because in cylindrical coordinates the required computational grid (the number of discrete points) is ∼10 times less comparing to the Cartesian coordinates. The less computational grid also leads to the 10 times reduction of the minimal required computational resources, especially the amount of GPU memory. As a result, the numerical simulation of the one-electron processes in largescale nanosystems (up to 103 nm3 ) can be performed on ordinary personal computer, equipped with modern GPU (Tesla k20 or higher). For the supercomputer calculations, the significant advantage is much better scalability: the GPU TDSE Solver shows linear scalability up to the 32 GPUs, that corresponds to the calculation domain about 3×104 nm3 (compare to the 3×103 nm3 for the previous version). The GPU TDSE Solver was applied to the calculation of the resonant charge transfer (RCT) in nanosystems. The calculated neutralization probability for Li+ ions impinging on the Ag(100) surface shows a good quantitative agreement with the experimental data. The further development of the GPU TDSE Solver consists in the construction of the 3D pseudopotentials in order to describe such 3D effects as RCT orientation dependence. After that, we are going to implement the new module for the calculation of the short laser pulses interaction with atomic/molecular electronic system.