A fast and accurate coupled meshless algorithm for the 2D/3D Gross-Pitaevskii equations on two GPUs

This paper first presents a high-efficient and accurate coupled meshless algorithm for solving the multi-dimensional Gross-Pitaevskii equation (GPE) in unbounded domain, which is implemented on CUDA-program-based two-GPUs cards. The proposed novel high-performance scheme (RDFPM-PML-GPU) is mainly motived by the items below: (a) a reduced-dimensional finite pointset method (RDFPM) is first presented to solve the 2D/3D spatial derivatives in GPE, which has lower calculated amount than the traditional FPM (TFPM) for the derivatives; (b) the perfectly matched layer (PML) technique is adopted to treat the absorbing boundary conditions (ABCs) which is used for the infinite exterior region, and the time-splitting technique is resorted to reduce the computing complexity in PML; (c) a fast parallel algorithm based on CUDA-program is proposed to accelerate the computation in the proposed meshless scheme with local matrix on two GPUs. The numerical convergent rate and advantages of the proposed meshless scheme are demonstrated by solving two examples, which include the comparisons between the proposed RDFPM and TFPM, the merit of easily implemented local refinement point distribution in meshless method, and the merit of PML-ABCs over the zero Dirichlet boundary treatment for unbounded domain. Meanwhile, the high efficiency of the proposed GPU-based parallelization algorithm is tested and discussed by simulating 3D examples, which shows that the speed-up rate is about 500-times of using two-GPUs over a single CPU. Finally, the proposed RDFPM-PML-GPU method is used to predict the long-time evolution of quantum vortex in 2D/3D GPEs describing Bose-Einstein condensates. All the numerical tests show the high-performance and flexible application of the proposed parallel meshless algorithm.