This paper is concerned with long-time strong approximations of SDEs with non-globally Lipschitz coefficients.Under certain non-globally Lipschitz conditions, a long-time version of fundamental strong convergence theorem is established for general one-step time discretization schemes. With the aid of the fundamental strong convergence theorem, we prove the expected strong convergence rate over infinite time for two types of schemes such as the backward Euler method and the projected Euler method in non-globally Lipschitz settings. Numerical examples are finally reported to confirm our findings.

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