  This paper presents a Crank-Nicolson leap-frog (CNLF) scheme for the unsteady
incompressible magnetohydrodynamics (MHD) equations. The spatial discretization
adopts the Galerkin finite element method (FEM), and the temporal
discretization employs the CNLF method for linear terms and the semi-implicit
method for nonlinear terms. The first step uses Stokes style's scheme, the
second step employs the Crank-Nicolson extrapolation scheme, and others apply
the CNLF scheme. We testify that the fully discrete scheme is stable and
convergent when the time step is less than or equal to a positive constant. The
second order $L^{2}$ error estimates can be derived by a novel negative norm
technique. The numerical results are consistent with our theoretical analysis,
which indicates that the method has an optimal convergence order. Therefore,
the scheme is effective for different parameters.
