  We consider weak convergence of one-step schemes for solving stochastic
differential equations (SDEs) with one-sided Lipschitz conditions. It is known
that the super-linear coefficients may lead to a blowup of moments of solutions
and their numerical solutions. When solutions to SDEs have all finite moments,
weak convergence of numerical schemes has been investigated in [Wang et al
(2023), Weak error analysis for strong approximation schemes of SDEs with
super-linear coefficients, IMA Journal numerical analysis]. Some modified Euler
schemes have been analyzed for weak convergence. In this work, we present a
family of explicit schemes of first and second-order weak convergence based on
classical schemes for SDEs. We explore the effects of limited moments on these
schemes. We provide a systematic but simple way to establish weak convergence
orders for schemes based on approximations/modifications of drift and diffusion
coefficients. We present several numerical examples of these schemes and show
their weak convergence orders.
