A discontinuous Galerkin time-domain (DGTD) method with a dynamic h-adaptation technique is presented in this paper. The adaptive Cartesian mesh (ACM) is employed so that the mesh elements get refined in real time of a simulation when the highly oscillatory electromagnetic pulse is passing by to achieve a desired resolution of the physics. When the pulse passes through or gets dissipated in the simulation domain, the mesh elements are automatically coarsened to reduce the computational cost. A local time-stepping scheme is adopted to alleviate the constraint on the time step size and maintain the numerical stability by advancing the fields in different elements with their own time step sizes. The proposed dynamic h-adaptation method with the local time-stepping scheme is extremely suitable for simulating the time-domain electromagnetic and multiphysics problems.