  We propose a Lawson-time-splitting extended Fourier pseudospectral (LTSeFP)
method for the numerical integration of the Gross-Pitaevskii equation with
time-dependent potential that is of low regularity in space. For the spatial
discretization of low regularity potential, we use an extended Fourier
pseudospectral (eFP) method, i.e., we compute the discrete Fourier transform of
the low regularity potential in an extended window. For the temporal
discretization, to efficiently implement the eFP method for time-dependent low
regularity potential, we combine the standard time-splitting method with a
Lawson-type exponential integrator to integrate potential and nonlinearity
differently. The LTSeFP method is both accurate and efficient: it achieves
first-order convergence in time and optimal-order convergence in space in
$L^2$-norm under low regularity potential, while the computational cost is
comparable to the standard time-splitting Fourier pseudospectral method.
Theoretically, we also prove such convergence orders for a large class of
spatially low regularity time-dependent potential. Extensive numerical results
are reported to confirm the error estimates and to demonstrate the superiority
of our method.
