  We present and analyze two numerical methods for the logarithmic
Schr{\"o}dinger equation (LogSE) consisting of a regularized splitting method
and a regularized conservative Crank-Nicolson finite difference method (CNFD).
In order to avoid numerical blow-up and/or to suppress round-off error due to
the logarithmic nonlinearity in the LogSE, a regularized logarithmic
Schr{\"o}dinger equation (RLogSE) with a small regularized parameter 0 <
$\epsilon$ $\ll$ 1 is adopted to approximate the LogSE with linear convergence
rate O($\epsilon$). Then we use the Lie-Trotter splitting integrator to solve
the RLogSE and establish its error bound O($\tau$ 1/2 ln($\epsilon$ --1)) with
$\tau$ > 0 the time step, which implies an error bound at O($\epsilon$ + $\tau$
1/2 ln($\epsilon$ --1)) for the LogSE by the Lie-Trotter splitting method. In
addition, the CNFD is also applied to discretize the RLogSE, which conserves
the mass and energy in the discretized level. Numerical results are reported to
confirm our error bounds and to demonstrate rich and complicated dynamics of
the LogSE.
