  As a classical time-stepping method, it is well-known that the Strang
splitting method reaches the first-order accuracy by losing two spatial
derivatives. In this paper, we propose a modified splitting method for the 1D
cubic nonlinear Schr\"odinger equation: \begin{align*}
u^{n+1}=\mathrm{e}^{i\frac\tau2\partial_x^2}{\mathcal N}_\tau
\left[\mathrm{e}^{i\frac\tau2\partial_x^2}\big(\Pi_\tau +\mathrm{e}^{-2\pi
i\lambda M_0\tau}\Pi^\tau \big)u^n\right], \end{align*} with ${\mathcal
N}_t(\phi)=\mathrm{e}^{-i\lambda t|\Pi_\tau\phi|^2}\phi,$ and $M_0$ is the mass
of the initial data. Suitably choosing the filters $\Pi_\tau$ and $\Pi^\tau$,
it is shown rigorously that it reaches the first-order accuracy by only losing
$\frac32$-spatial derivatives. Moreover, if $\gamma\in (0,1)$, the new method
presents the convergence rate of $\tau^{\frac{4\gamma}{4+\gamma}}$ in
$L^2$-norm for the $H^\gamma$-data; if $\gamma\in [1,2]$, it presents the
convergence rate of $\tau^{\frac25(1+\gamma)-}$ in $L^2$-norm for the
$H^\gamma$-data. %In particular, the regularity requirement of the initial data
for the first-order convergence in $L^2$-norm is only $H^{\frac32+}$. These
results are better than the expected ones for the standard (filtered) Strang
splitting methods. Moreover, the mass is conserved: $$\frac1{2\pi}\int_{\mathbb
T} |u^n(x)|^2\,d x\equiv M_0, \quad n=0,1,\ldots, L . $$ The key idea is based
on the observation that the low frequency and high frequency components of
solutions are almost separated (up to some smooth components). Then the
algorithm is constructed by tracking the solution behavior at the low and high
frequency components separately.
