  Yen et al. (2012) advanced a direct approach for the calculation of
self-gravitational force to second order accuracy based on uniform grid
discretization. This method improves the accuracy of N-body calculation by
using exact integration of kernel functions and employing the Fast Fourier
Transform (FFT) to reduce complexity of computation to nearly linear. This
direct approach is free of artificial boundary conditions, however, the
applicability is limited by the uniform discretization of grids. We report here
an advancement in the direct method with the implementation of adaptive mesh
refinement (AMR) and maintaining second-order accuracy, which breaks the
barrier set by uniform grid discretization. The adoption of graphic process
units (GPUs) can significantly speed up the computation and make application of
this method possible to astrophysical systems of gaseous disk galaxies and
protoplanetary disks.
