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This paper presents the study of coupling efficiencies between two-dimensional (2-D) waveguides and microring resonators with a newly developed high-order discontinuous Galerkin time domain (DGTD) method for Maxwell's equations. The DGTD method is based on a unified formulation for the physical media and the artificial media in the uniaxial perfectly matched layer (UPML) regions used to truncate the computational domain. The DGTD method employs finite element type meshes and uses piecewise high-order polynomials for spatial discretization of the Maxwell's equations and Runge-Kutta methods for time integration. After demonstrating the high-order convergence of the DGTD method, the effect of separation gap between the waveguides and one and two microrings on the coupling efficiency and transmittance for pulse propagations is studied.