The dynamic simulation of the integrated electricity and gas system (IEGS) is effective for accurate operational guidance. However, the IEGS model governed by partial and ordinary differential-algebraic equations (PDAE) is challenging for analysis, because it poses heterogeneous properties, high dimensionality, and strong nonlinearity. To eliminate the nonlinear terms, this paper firstly derives the expression of average flow velocity (AFV) and reformulates the gas flow dynamics into the variable-coefficient partial differential equations (VC-PDE), whose consistency, stability, and convergence can be theoretically guaranteed. Then, a novel adaptive step size control (ASC) strategy is developed for VC-PDE solvers to improve computational efficiency and accuracy. Finally, this paper builds a framework for the simulation of IEGS considering the dynamics in the natural gas network (NGN) and coupling units, where the bidirectional interaction between the electric power system (EPS) and NGN is investigated. Case studies verify the superiority of the proposed VC-PDE solver with ASC regarding accuracy, stability, convergence, and efficiency in different systems.