An explicit marching-on-in-time (MOT) scheme to efficiently solve the time domain magnetic field integral equation (TD-MFIE) with a large time step size (under a low-frequency excitation) is developed. The proposed scheme spatially expands the current using high-order nodal functions defined on curvilinear triangles discretizing the scatterer surface. Applying Nyström discretization, which uses this expansion, to the TD-MFIE, which is written as an ordinary differential equation (ODE) by separating self-term contribution, yields a system of ODEs in unknown time-dependent expansion coefficients. A predictor-corrector method is used to integrate this system for samples of these coefficients. Since the Gram matrix arising from the Nyström discretization is blockdiagonal, the resulting MOT scheme replaces the matrix “inversion” required at each time step by a product of the inverse block-diagonal Gram matrix and the right-hand side vector. It is shown that, upon convergence of the corrector updates, this explicit MOT scheme produces the same solution as its implicit counterpart, and is faster for large time step sizes.