Due to the development of the third-generation semiconductors representative of gallium nitride (GaN), the microwave power devices are developing towards higher power, higher efficiency and high integration. However, the electromagnetic field effects are more significant inside the device. As a result, circuit-level based simulation techniques can no longer satisfy the accuracy requirements of device design. Therefore it is necessary to urgently establish the field-circuit co-simulation techniques to couple the active GaN devices with passive electromagnetic structures. In this work, we propose a high-precision discontinuous Galerkin time-domain method to analyze the performances of GAN-based high-power microwave devices. The extracted large-signal compact model of the GaN HEMT is incorporated into the electromagnetic field equations. A local time-stepping technique is adopted to remove the constraints of nonlinear compact models and multiscale elements on the stability conditions of the global algorithm. The comparisons among numerical simulations, experimental results, and software calculations demonstrate the excellent accuracy and efficiency of the proposed method, which can provide a theoretical analysis and design tool for the high reliability design of advanced high-power microwave devices.