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The root cause of the instability is quantitatively identified for the explicit time-domain finite-element method that employs a time step beyond that allowed by the stability criterion. With the identification of the root cause, an unconditionally stable explicit time-domain finite-element method is successfully created, which is stable and accurate for a time step solely determined by accuracy regardless of how large the time step is. The proposed method retains the strength of an explicit time-domain method in avoiding solving a matrix equation while eliminating its shortcoming in time step. Numerical experiments have demonstrated its superior performance in computational efficiency, as well as stability, compared with the conditionally stable explicit method and the unconditionally stable implicit method. The essential idea of the proposed method for making an explicit method stable for an arbitrarily large time step irrespective of space step is also applicable to other time domain methods.