## 2 Basic Theory
### Common Approach of TDG Methods
#### 2.1.1 Differences Between the Conventional TDG and the Optimized TDG
Especially in terms of computational accuracy, to maintain the computational accuracy of the center difference, then at the boundary, it is necessary to construct a difference form that covers the information of multiple cells. In the optimized TDG method, the nodal DG approach is utilized to discretize elements into a set of discrete points. Based on this discretization, the information at these points can be leveraged to construct new differential forms. These differential forms are primarily required for estimating first-order spatial derivatives. Compared to the modal DG method, the nodal DG approach provides more detailed information within each element, eliminating the need to rely on data from neighboring elements. If the central difference method is used, the difference formulation within an element is influenced by the positions of the discrete points. This necessitates special handling for the discrete points located at the edges of the element. To address this, the present study employs a higher-order Taylor expansion to construct a differential structure that effectively utilizes the available information at these points. This approach ensures an organized computational framework, where the coefficients of the differential formulation can be precomputed based on the positions of the discrete points. This precomputation is performed while the mesh remains constant, prior to starting the time iteration process. This Taylor difference method estimates the first-order spatial derivatives of unknowns at discrete points by employing the estimated values of all discrete points in one element, as in \[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]5 The process in the one-dimensional format is as follows:  
1. Discretization: A one-dimensional segment is treated as a computational element, which is discretized into N points. The estimated values of the governing equations at these points are known;
2. Taylor expansion: For each point \(x_{i}\) in the element, the function values at all other points \(x_{j}(j=1,2,\ldots,N)\) are expressed using a Taylor series expansion around \(x_{i}\): \[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]6 where \(f^{\prime}(x_{i})\) and higher-order derivatives are unknown;
3. Equation system: Using the known function values \(f(x_{i})\) at the discrete points, the Taylor expansions for all N points are combined to form a system of N equations;
4. Solution: By solving this system of equations, the first-order spatial derivative \(f^{\prime}(x_{i})\) is computed for each point \(x_{i}\). Higher-order derivatives may also be obtained if needed, but only the first-order derivative is typically retained in this algorithm;
5. Output: The algorithm produces the first-order spatial derivative \(f^{\prime}(x_{i})\) for each point \(x_{i}\) in the element and the coefficients \(a_{ij}(x_{i})\) of the N equations system. The finite difference function \(TD(x_{i})\) is: \[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]7The above computations correspond to the process of determining finite difference coefficients based on the positions of discrete points.