Non-asymptotic Error Analysis of Explicit Modified Euler Methods for Superlinear and Non-contractive SODEs

Abstract

A family of explicit modified Euler methods (MEMs) is constructed for long-time approximations of super-linear SODEs driven by multiplicative noise. The proposed schemes can preserve the same Lyapunov structure as the continuous problems. Under a non-contractive condition, we establish a non-asymptotic error bound between the law of the numerical approximation and the target distribution in Wasserstein-1 (${\mathcal{W}}_1$) distance through a time-independent weak convergence rate for the proposed schemes. As a by-product of this weak error estimate, we obtain an $\mathcal{O}(\ddot{I}„|\ln \ddot{I}„|)$ convergence rate between the exact and numerical invariant measures.

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