  We present new high order approximations schemes for the Cox-Ingersoll-Ross
(CIR) process that are obtained by using a recent technique developed by
Alfonsi and Bally (2021) for the approximation of semigroups. The idea consists
in using a suitable combination of discretization schemes calculated on
different random grids to increase the order of convergence. This technique
coupled with the second order scheme proposed by Alfonsi (2010) for the CIR
leads to weak approximations of order $2k$, for all $k\in\mathbb{N}^*$. Despite
the singularity of the square-root volatility coefficient, we show rigorously
this order of convergence under some restrictions on the volatility parameters.
We illustrate numerically the convergence of these approximations for the CIR
process and for the Heston stochastic volatility model and show the
computational time gain they give.
