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We present 2D MHD numerical simulations of tearing-unstable current sheetscoupled to a population of non-thermal test-particles, in order to address theproblem of numerical convergence with respect to grid resolution, numericalmethod and physical resistivity. Numerical simulations are performed with thePLUTO code for astrophysical fluid dynamics through different combinations ofRiemann solvers, reconstruction methods, grid resolutions at various Lundquistnumbers. The constrained transport method is employed to control thedivergence-free condition of magnetic field. Our results indicate that thereconnection rate of the background tearing-unstable plasma converges only forfinite values of the Lundquist number and for sufficiently large gridresolutions. In general, it is found that (for a 2nd-order scheme) the minimumthreshold for numerical convergence during the linear phases requires thenumber of computational zones covering the initial current sheet width to scaleroughly as $\sim \sqrt{\bar{S}}$, where $\bar{S}$ is the Lundquist numberdefined on the current sheet width. On the other hand, the process of particleacceleration is found to be nearly independent of the underlying numericaldetails inasmuch as the system becomes tearing-unstable and enters in itsnonlinear stages. In the limit of large $\bar{S}$, the ensuing power-law indexquickly converge to $p \approx 1.7$, consistently with the fast reconnectionregime.

On the Impact of the Numerical Method on Magnetic Reconnection and Particle Acceleration -- I. The MHD case