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Dynamic simulation plays a fundamental role in security evaluation of distribution networks (DNs). However, the strong stiffness and non-linearity of distributed generation (DG) models in DNs bring about burdensome computation and noteworthy instability on traditional methods which hampers the rapid response of simulation tool. Thus, a novel L-stable approximate analytical method with high accuracy is proposed to handle these problems. The method referred to as multi-stage discontinuous Galerkin method (MDGM), first derives approximate analytical solutions (AASs) of state variables which are explicit symbolic expressions concerning system states. Then, in each time window, it substitutes values for symbolic variables and trajectories of state variables are obtained subsequently. This paper applies MDGM to DG models to derive AASs. Local-truncation-error-based variable step size strategy is also developed to further improve simulation efficiency. In addition, this paper establishes detailed MDGM-based dynamic simulation procedure. From case studies on a numerical problem, a modified 33-bus system and a practical large-scale DN, it can be seen that proposed method demonstrates fast and dependable performance compared with the traditional trapezoidal method.