  For semilinear stochastic evolution equations whose coefficients are more
general than the classical global Lipschitz, we present results on the strong
convergence rates of numerical discretizations. The proof of them provides a
new approach to strong convergence analysis of numerical discretizations for a
large family of second order parabolic stochastic partial differential
equations driven by space-time white noises. We apply these results to the
stochastic advection-diffusion-reaction equation with a gradient term and
multiplicative white noise, and show that the strong convergence rate of a
fully discrete scheme constructed by spectral Galerkin approximation and
explicit exponential integrator is exactly $\frac12$ in space and $\frac14$ in
time. Compared with the optimal regularity of the mild solution, it indicates
that the spetral Galerkin approximation is superconvergent and the convergence
rate of the exponential integrator is optimal. Numerical experiments support
our theoretical analysis.
