## 2 Basic Theory
### Common Approach of TDG Methods
#### 2.1.1 Differences Between the Conventional TDG and the Optimized TDG
Although the nodal DG method is employed for spatial discretization in this study, the FDM is applied to the discrete nodes. The discrete nodes and elements constructed by the nodal DG method are illustrated in Figure 2.  
When the nodal DG method is used, an estimated solution \(\mathbf{u}_{(k,i)}^{n}\) and an inertia term for a discrete point \(i\) in an element \(K\) at a time \(n\) can be obtained directly by calculating the spatial integral. Consequently, the value of matrix \(\mathbf{\Lambda}_{(k,i)}^{n}\) at a point \(i\) can be obtained. According to Equations (15) and (16), functions \(G_{(k,i)}^{n}\) and \(H_{(k,i)}^{n}\) are derived as follows:  
\[G_{(k,i)}^{n}=\left.\mathbf{\Lambda}_{(k,i)}^{n}\frac{\partial\mathbf{u}_{(k,i)}^{n}(x )}{\partial t}\right|_{t=n}, \tag{20}\]  
\[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]0  
where \(n\), \(k\), and \(i\) denote the time, element number, and position number of a point in an element, respectively. In this study, the fourth-order central difference approximation is used to calculate the high-order temporal derivatives by outside element information. As illustrated in Figure 3, to calculate the high-order temporal derivatives at a target point i in an element K, the values of functions G and H at point i in four elements surrounding the target point must be obtained. The difference form of the second-order temporal derivative term is calculated by:  
\[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]1  
In the same scheme, the different form of the third-order temporal derivative term is obtained by:  
\[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]2  
These operations are used when the optimized method is applied to the test cases presented in this study, which is called the optimized TDG outside element method. \[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]3 The illustration of the value position of a function G by difference method inside element. \[\overline{\mathbf{u}}_{ttt}^{n}=\left.-\frac{\partial}{\partial x}\Bigg{(}\frac{ \partial\mathbf{A}}{\partial\mathbf{u}}\bigg{(}\frac{\partial\mathbf{f}}{\partial x} \bigg{)}^{2}+\mathbf{A}\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{ \partial\mathbf{f}}{\partial x}\bigg{)}\Bigg{)}\right|_{t=n}. \tag{12}\]4 The illustration of the value position of a function G by difference method among elements. 2.1.3. The Calculation of High-Order Temporal Derivatives by the Finite Difference Method Inside the Element  
There will be a distinct disadvantage in the above approach, which is that at the boundaries, it is not possible to construct the center difference format directly. At the edges of the computational domain, it is necessary to use one-side difference form. This problem is relatively easy to solve in one-dimensional problems, but for two-dimensional problems it requires a lot of effort to construct the difference form at the boundary.