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In this paper, a novel high-order accurate numerical method was proposed to simulate the space charge transport in solid dielectrics. It is based on a unified framework, discontinuous Galerkin (DG) method, to solve both charge concentration and electric field. It is easier to implement and more efficient than the combined DG and boundary element method (BEM) with equivalent numerical accuracy. With this new method and symmetric model parameters for low-density polyethylene (LDPE), space charge concentration and electric field distribution under different external electric field and injection barrier heights were obtained. The simulation shows that above an electric field threshold, large oscillations appear in the time-dependent maximum electric field strength and the total charge quantity. This phenomenon can be considered as a sign of a relatively larger quantity of charge injection compared to the low field regime. When the Schottky injection barrier height in the simulation is raised, this electric field threshold also increases in order to maintain a certain amount of carrier injection.