In typical stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one- dimensional stochastic differential equation (SDE). We assume that the coefficients of this equation are smooth. Using Ito's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose first a scheme based on the Milstein discretization of this SDE, which converges with order 1 to the asset price dynamics for an appropriate notion of convergence that we call weak trajectorial convergence, and, second, a scheme based on the Ninomiya-Victoir discretization of this SDE, with order 2, of weak convergence to the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Ornstein-Uhlenbeck process.We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles in 2008.