  Let $(X_t)_{t \ge 0}$ be the solution of the stochastic differential equation
$$dX_t = b(X_t) dt+A dZ_t, \quad X_{0}=x,$$ where $b: \mathbb{R}^d \rightarrow
\mathbb R^d$ is a Lipschitz function, $A \in \mathbb R^{d \times d}$ is a
positive definite matrix, $(Z_t)_{t\geq 0}$ is a $d$-dimensional rotationally
invariant $\alpha$-stable L\'evy process with $\alpha \in (1,2)$ and
$x\in\mathbb{R}^{d}$. We use two Euler-Maruyama schemes with decreasing step
sizes $\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant
measure of $(X_t)_{t \ge 0}$: one with i.i.d. $\alpha$-stable distributed
random variables as its innovations and the other with i.i.d. Pareto
distributed random variables as its innovations. We study the convergence rate
of these two approximation schemes in the Wasserstein-1 distance. For the first
scheme, when the function $b$ is Lipschitz and satisfies a certain dissipation
condition, we show that the convergence rate is $\gamma^{1/\alpha}_n$. Under an
additional assumption on the second order directional derivatives of $b$, this
convergence rate can be improved to $\gamma^{1+\frac 1
{\alpha}-\frac{1}{\kappa}}_n$ for any $\kappa \in [1,\alpha)$. For the second
scheme, when the function $b$ is twice continuously differentiable, we obtain a
convergence rate of $\gamma^{\frac{2-\alpha}{\alpha}}_n$. We show that the rate
$\gamma^{\frac{2-\alpha}{\alpha}}_n$ is optimal for the one dimensional stable
Ornstein-Uhlenbeck process. Our theorems indicate that the recent remarkable
result about the unadjusted Langevin algorithm with additive innovations can be
extended to the SDEs driven by an $\alpha$-stable L\'evy process and the
corresponding convergence rate has a similar behaviour. Compared with the
previous result, we have relaxed the second order differentiability condition
to the Lipschitz condition for the first scheme.
