  The present work introduces and investigates an explicit time discretization
scheme, called the projected Euler method, to numerically approximate random
periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions.
The existence of the random periodic solution is demonstrated as the limit of
the pull-back of the discretized SDE. Without relying on a priori high-order
moment bounds of the numerical approximations, the mean square convergence rate
is proved to be order 0.5 for SDEs with multiplicative noise and order 1 for
SDEs with additive noise. Numerical examples are also provided to validate our
theoretical findings.
