  We propose and analyse boundary-preserving schemes for the strong
approximations of some scalar SDEs with non-globally Lipschitz drift and
diffusion coefficients whose state-space is bounded. The schemes consists of a
Lamperti transform followed by a Lie--Trotter splitting. We prove
$L^{p}(\Omega)$-convergence of order $1$, for every $p \geq 1$, of the schemes
and exploit the Lamperti transform to confine the numerical approximations to
the state-space of the considered SDE. We provide numerical experiments that
confirm the theoretical results and compare the proposed Lamperti-splitting
schemes to other numerical schemes for SDEs.
