  In this article, we consider numerical schemes for polynomial diffusions on
the unit ball, which are solutions of stochastic differential equations with a
diffusion coefficient of the form $\sqrt{1-|x|^{2}}$. We introduce a
semi-implicit Euler--Maruyama scheme with the projection onto the unit ball and
provide the $L^{2}$-rate of convergence. The main idea to consider the
numerical scheme is the transformation argument introduced by Swart for proving
the pathwise uniqueness for some stochastic differential equation with a
non-Lipschitz diffusion coefficient.
