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A high-precision integration (HPI) scheme combined with the spectral-element time-domain (SETD) method is presented to solve the time-dependent Maxwell's equations. Spatial discretization by spectral elements will lead to block diagonal mass matrices, thus greatly alleviating the computational burden of inverting the mass matrices. A high-precision time integration method based on the matrix exponential is then employed to solve the discretized SETD system. Numerical examples demonstrate that this algorithm is unconditionally stable and very accurate.