We performed numerical simulations of general relativistic magnetohydrodynamics with uniform resistivity to investigatethe occurrence of magnetic reconnection in a split-monopole magnetic field around a Schwarzschild black hole. We found that magnetic reconnection happens near the black hole at its equatorial plane... The magnetic reconnection has a point-like reconnection region and slow shock waves, as in the Petschek reconnection model. The magnetic reconnection rate decreases as the resistivity becomes smaller. When the global magnetic Reynolds number is $10^4$ or larger, the magnetic reconnection rate increases linearly with time from $2 \tau_{\rm S}$ to $\sim 10 \tau_{\rm S}$ ($\tau_{\rm S}=r_{\rm S}/c, r_{\rm S}$ is the Schwarzschild radius and $c$ is the speed of light). The linear increase of the reconnection rate agrees with the magnetic reconnection in the Rutherford regime of the tearing mode instability. (read more)