Ergodic Estimates of One-Step Numerical Approximations for Superlinear SODEs

This paper establishes the first-order convergence rate for the ergodic error of numerical approximations to a class of stochastic ODEs (SODEs) with superlinear coefficients and multiplicative noise. By leveraging the generator approach to the Stein method, we derive a general error representation formula for one-step numerical schemes. Under suitable dissipativity and smoothness conditions, we prove that the error between the accurate invariant measure $\pi$ and the numerical invariant measure $\pi_\tau$ is of order $\mathscr{O}(\tau)$, which is sharp. Our framework applies to several recently studied schemes, including the tamed Euler, projected Euler, and backward Euler methods.