New general approach in few-body scattering calculations: Solving discretized Faddeev equations on a graphics processing unit

Background: The numerical solution of few-body scattering problems with realistic interactions is a difficult problem that normally must be solved on powerful supercomputers, taking a lot of computer time. This strongly limits the possibility of accurate treatments for many important few-particle problems in different branches of quantum physics.

Purpose: To develop a new general highly effective approach for the practical solution of few-body scattering equations that can be implemented on a graphics processing unit.

Methods: The general approach is realized in three steps: (i) the reformulation of the scattering equations using a convenient analytical form for the channel resolvent operator, (ii) a complete few-body continuum discretization and projection of all operators and wave functions onto an $L_2$ basis constructed from stationary wave packets, and (iii) the ultrafast solution of the resulting matrix equations using a graphics processor.

Results: The whole approach is illustrated by a calculation of the neutron-deuteron elastic scattering cross section below and above the three-body breakup threshold with a realistic $NN$ potential that is performed on a standard PC using a graphics processor with an extremely short runtime.

Conclusions: The general technique proposed in this paper opens a new way for a fast practical solution of quantum few-body scattering problems in both nonrelativistic and relativistic formulations in hadronic, nuclear, and atomic physics.

Figure 1

Some of $S$- and $D$-wave partial phase shifts of the elastic $nd$ scattering obtained within the WP approach (solid lines) and within the standard Faddeev calculations (circles) [11].

Figure 2

The differential cross section for elastic $nd$ scattering at 13 MeV obtained via the WP technique (solid line) and within the standard Faddeev calculations [11] (dashed line).

Figure 3

The neutron vector analyzing power $A_y$ for the elastic $nd$ scattering at 35 MeV obtained within the WP approach (solid line) compared to the results from Ref. [11] (dashed line) and the experimental $pd$ data [17].

Figure 4

The dependence of the computing time on a CPU (solid lines) and a GPU (dashed lines) (for the full solution and separate steps) on the dimension of the basis $M=N$ for the $nd$ scattering problem with $S$-wave $NN$ interaction.

Figure 5

The dependence of the GPU-acceleration ratio $\eta$ on the dimension of the basis $M=N$ in the calculation of the $nd$ scattering problem with an $S$-wave $NN$ interaction. The dashed line shows the acceleration for step 2 (selection), the dot-dashed line shows the acceleration for step 3 (calculation of ${\mathbb{P}}^0$), and the solid line shows the acceleration for the complete solution.

Figure 6

The dependence of the GPU acceleration ratio $\eta$ on the dimension of the basis $M=N$ for the realistic $nd$ scattering problem for $J={\frac{1}{2}}^{+}$. The dashed line shows the acceleration for step 2 (selection), the dot-dashed line shows the acceleration for step 3 (calculation of ${\mathbb{P}}^0$), and the solid line shows the acceleration for the complete solution.