  In this work, we develop a numerical method to study the error estimates of
the $\alpha$-stable central limit theorem under sublinear expectation with
$\alpha \in(0,2)$, whose limit distribution can be characterized by a fully
nonlinear integro-differential equation (PIDE). Based on the sequence of
independent random variables, we propose a discrete approximation scheme for
the fully nonlinear PIDE. With the help of the nonlinear stochastic analysis
techniques and numerical analysis tools, we establish the error bounds for the
discrete approximation scheme, which in turn provides a general error bound for
the robust $\alpha$-stable central limit theorem, including the integrable case
$\alpha \in(1,2)$ as well as the non-integrable case $\alpha \in(0,1]$.
Finally, we provide some concrete examples to illustrate our main results and
derive the precise convergence rates.
