  Sampling with Markov chain Monte Carlo methods often amounts to discretizing
some continuous-time dynamics with numerical integration. In this paper, we
establish the convergence rate of sampling algorithms obtained by discretizing
smooth It\^o diffusions exhibiting fast Wasserstein-$2$ contraction, based on
local deviation properties of the integration scheme. In particular, we study a
sampling algorithm constructed by discretizing the overdamped Langevin
diffusion with the method of stochastic Runge-Kutta. For strongly convex
potentials that are smooth up to a certain order, its iterates converge to the
target distribution in $2$-Wasserstein distance in
$\tilde{\mathcal{O}}(d\epsilon^{-2/3})$ iterations. This improves upon the
best-known rate for strongly log-concave sampling based on the overdamped
Langevin equation using only the gradient oracle without adjustment. In
addition, we extend our analysis of stochastic Runge-Kutta methods to uniformly
dissipative diffusions with possibly non-convex potentials and show they
achieve better rates compared to the Euler-Maruyama scheme in terms of the
dependence on tolerance $\epsilon$. Numerical studies show that these
algorithms lead to better stability and lower asymptotic errors.
