  In this paper, we investigate the damped stochastic nonlinear
Schr\"odinger(NLS) equation with multiplicative noise and its splitting-based
approximation. When the damped effect is large enough, we prove that the
solutions of the damped stochastic NLS equation and the splitting scheme are
exponential stable and possess some exponential integrability.
  These properties lead that the strong order of the scheme is $\frac 12$ and
independent of time. Meanwhile, we analyze the regularity of the Kolmogorov
equation with respect to the equation. As a consequence, the weak order of the
scheme is shown to be twice the strong order and independent of time.
