  We present 2D MHD numerical simulations of tearing-unstable current sheets
coupled to a population of non-thermal test-particles, in order to address the
problem of numerical convergence with respect to grid resolution, numerical
method and physical resistivity. Numerical simulations are performed with the
PLUTO code for astrophysical fluid dynamics through different combinations of
Riemann solvers, reconstruction methods, grid resolutions at various Lundquist
numbers. The constrained transport method is employed to control the
divergence-free condition of magnetic field. Our results indicate that the
reconnection rate of the background tearing-unstable plasma converges only for
finite values of the Lundquist number and for sufficiently large grid
resolutions. In general, it is found that (for a 2nd-order scheme) the minimum
threshold for numerical convergence during the linear phases requires the
number of computational zones covering the initial current sheet width to scale
roughly as $\sim \sqrt{\bar{S}}$, where $\bar{S}$ is the Lundquist number
defined on the current sheet width. On the other hand, the process of particle
acceleration is found to be nearly independent of the underlying numerical
details inasmuch as the system becomes tearing-unstable and enters in its
nonlinear stages. In the limit of large $\bar{S}$, the ensuing power-law index
quickly converge to $p \approx 1.7$, consistently with the fast reconnection
regime.
