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.

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