## 1 Introduction  
The Euler equations, fundamental to computational fluid dynamics, require high-order temporal accuracy to capture intricate flow features in various engineering and scientific applications. Among numerical methods, the Discontinuous Galerkin (DG) method is particularly notable. It extends the principles of the finite element method (FEM) to offer flexibility for complex geometries, efficient parallelization, and the ability to handle discontinuities at element interfaces [1, 2]. Initially developed by [PERSON] and [PERSON] [1] for the neutron transport equation, the DG method was subsequently adapted for polynomial-based approximations [2]. [PERSON] and colleagues [3, 4, 5] extended the DG framework to time-dependent problems, integrating the Runge-Kutta (RK) method into the DG framework to establish the RKDG method [6]. This powerful combination refined the approach for conservation laws and ensured convergence, facilitating simulations across diverse fields. In a recent study [7], the phenomenon of the interaction of shock waves with forward-triangular light gas bubbles was simulated using the high-fidelity explicit modal discontinuous Galerkin technique. This work demonstrates that the DG method provides excellent simulation capabilities for complex aerodynamic phenomena by solving the Eulerequations. Additionally, this simulation enables in-depth exploration of the impact of different parameter conditions on flow morphology through comparison and observation. The above work demonstrates that using numerical computation methods based on the DG method for simulating fluid problems governed by the Euler equations as the governing equations can obtain excellent results. The DG method is primarily employed for spatial discretization in time-dependent conservative partial differential equations (PDEs). For temporal discretization, the RK method has become the standard for iteratively calculating temporal derivatives to achieve high-order accuracy. By contrast, the Taylor Discontinuous Galerkin (TDG) method offers an innovative alternative. Instead of relying on multiple low-order predictions typical of RK-based schemes, it directly employs high-order Taylor series expansions derived from the Lax-Wendroff-type time integration scheme [8]. This approach constructs high-order approximations without the need for iterative corrections, which can be computationally expensive, particularly in large-scale (advantage in CPU timing for the same accuracy by TDG) complex systems where the dimensionality and complexity significantly increase computational costs [9]. The potential of Taylor-Galerkin methods was first demonstrated by [PERSON] [10] in solving convective transport problems. Subsequent studies extended their application to incompressible flows involving heat transfer [11] and hyperbolic conservation laws [8]. Its semi-implicit adaptations have further widened its applicability. [PERSON] [12] highlighted the compactness of the TDG method and its significant time-saving advantages, making it particularly well suited for large-scale simulations. By avoiding the iterative procedures inherent in RK-based schemes, the TDG framework provides a direct and efficient path to achieving high-order temporal accuracy, advancing both computational efficiency and precision in solving Euler equations for large-scale complex phenomena. Nevertheless, implementing TDG methods for multi-dimensional and large-scale flow problems remains challenging. A key difficulty lies in evaluating the complex third-order tensor derivatives required by standard TDG formulations, which can lead to substantial computational overhead. Although recent modifications that incorporate central difference schemes have helped simplify the computation of high-order temporal derivatives by employing the finite difference method [12], these approaches often demand extensive information from neighboring elements, reducing their efficiency near domain boundaries.