## 1 Introduction
Moreover, while one-dimensional tests have successfully validated TDG's fundamental principles, many practical engineering applications--such as turbulent aeroacoustics, high-speed compressible flows, and multiphase problems--are inherently multi-dimensional and demand more efficient, robust approaches [8; 12]. This study addresses these issues by introducing an optimized TDG method that significantly reduces the computational burden of handling high-order derivatives. Instead of directly computing cumbersome third-order tensors, our approach employs temporal derivatives of the Jacobian matrix, thus simplifying the temporal differentiation process and reducing the number of neighboring elements needed for spatial approximations. After confirming the feasibility and accuracy of this optimized approach in one-dimensional settings, we apply it to a two-dimensional forward-facing step problem--a benchmark known for its complexity. The numerical results demonstrate that our optimized TDG method preserves the accuracy and stability expected from TDG techniques while markedly improving computational efficiency. Although the present work focuses on two-dimensional Euler equations, the fundamental concept can be naturally extended to three-dimensional domains, complex boundary conditions, and even coupled multi-physics problems, thereby offering a promising pathway for simulations for large-scale complex systems. In conventional TDG methods, the high-order construction makes the spatial integration in the DG method extremely complicated and involves the calculation of exact fluxes, which makes it difficult for the efficiency advantage of traditional TDG algorithms to be realized in complex flux problems. In [13], the integral terms were simplified by using high-order difference approximations in each element, which improved computational efficiency while ensuring accuracy. This indicates that using high-order difference methods within elements does not affect the computational accuracy of the TDG method and can enhance computational efficiency. In this study, the approach of using finite high-order difference methods to simplify calculations is considered in Section 3. Other researchers have also focused on simplifying the flux computation and the finite difference form of unknowns. In [14], the TDG method achieved sharper solution profiles near non-smooth areas by locally using finite difference methods to simplify flux computation and adjusting the differential parameters of the unknowns. However, the above improvements do not prevent the emergence of high-order matrix operators (like tensors). In [15], it was proven that the numerical computation method based on the Taylor expansion of the Lax-Wendroff temporal discretization combined with the DG method can ensure that the computational process of the Euler method maintains the requirement of positivity-preserving under a third-order expansion. This provides a feasible expectation for using the high-order expansion method of TDG to compute the Euler equations. The implementation details and numerical experiments presented here build upon the methods described in [16]. Following an initial demonstration of how our optimized TDG approach maintains high-order temporal accuracy and simplifies the weak form calculation for flux terms, we will discuss the theoretical formulation in detail, highlight the key modifications introduced, and present comprehensive numerical evidence of the method's effectiveness. Through these advancements, we aim to refine the TDG framework into a powerful, efficient, and versatile tool for large-scale multi-physics phenomena.