## 2 Basic Theory
### Common Approach of TDG Methods
#### 2.1.1 Differences Between the Conventional TDG and the Optimized TDG
To avoid the usage of \(\partial\mathbf{A}/\partial\mathbf{u}\), in this work, the high-order temporal derivatives are expressed as:  
\[\frac{\partial^{2}\mathbf{u}}{\partial t^{2}}=\frac{\partial}{\partial t}\bigg{(} \frac{\partial\mathbf{u}}{\partial t}\bigg{)}=\frac{\partial}{\partial t}\bigg{(} -\frac{\partial f}{\partial x}\bigg{)}=-\frac{\partial}{\partial x}\bigg{(} \mathbf{A}\frac{\partial\mathbf{u}}{\partial t}\bigg{)}, \tag{13}\]  
\[\frac{\partial^{3}\mathbf{u}}{\partial t^{3}}=\frac{\partial}{\partial t}\bigg{(} \frac{\partial^{2}\mathbf{u}}{\partial t^{2}}\bigg{)}=\frac{\partial}{\partial t} \bigg{(}-\frac{\partial}{\partial x}\bigg{(}\mathbf{A}\frac{\partial\mathbf{u}}{ \partial t}\bigg{)}\bigg{)}=-\frac{\partial}{\partial x}\bigg{(}\mathbf{A} \frac{\partial^{2}\mathbf{u}}{\partial t^{2}}+\frac{\partial A}{\partial t}\frac{ \partial\mathbf{u}}{\partial t}\bigg{)}, \tag{14}\]  
According to Equations (13) and (14), at the time \(=n\), the estimated value of the high-order temporal derivatives can be defined as:  
\[\overline{\mathbf{u}}_{tt}^{n}=\left.-\frac{\partial}{\partial x}\bigg{(}\mathbf{A} \frac{\partial\mathbf{u}}{\partial t^{2}}+\frac{\partial\mathbf{A}}{\partial t} \frac{\partial\mathbf{u}}{\partial t}\bigg{)}\right|_{t=n}. \tag{15}\]In this work, these new formulations with Equations (15) and (16) utilize \(\mathrm{A}_{t}=\partial\mathbf{A}/\partial t\) instead of \(\mathrm{A}_{u}=\partial\mathbf{A}/\partial u\), effectively avoiding the computational cost associated with high-order tensor operations while maintaining high temporal accuracy. Figure 1 illustrates difference between \(\mathbf{A}^{n}\), \(\mathbf{A}_{t}^{n}\), and tensor \(\mathbf{A}_{u}^{n}\) at time \(n\). This figure highlights how the proposed method simplifies tensor operations, reducing computational overhead. Obviously, to complete the calculations for Equations (15) and (16), we first need to determine the values of \(\partial u/\partial t\) and \(\partial\mathbf{A}/\partial t\). Based on Equation (4), the value of \(\partial u/\partial t\) can be obtained as follows:  
\[\frac{\partial u}{\partial t}=\int_{x^{(k-1)}}^{x^{(k)}}\frac{\partial\Phi}{ \partial x}\overline{\mu}^{n}dx-\left.\Phi\overline{\mu}_{x}^{n}\right|_{x^{( k-1)}}^{x^{(k)}}. \tag{17}\]  
Since the two-dimensional matrix \(\mathbf{A}_{t}\) can be expressed in terms of \(\mathbf{u}\) and \(\partial\mathbf{u}/\partial t\), the next equation is written as follows:  
\[\mathbf{A}_{t}=\mathrm{A}_{t}(\mathbf{u},\ \partial\mathbf{u}/\partial t). \tag{18}\]  
From this discussion, it is evident that the computational procedure for the optimized TDG method proposed in this work differs from that of the conventional TDG method, which typically begins with a Lax-Wendroff-type time integration scheme and concludes with spatial integration (via the weak formulation of the DG method). In contrast, the optimized TDG method first employs the DG method to perform spatial integration and thus obtain \(\partial\mathbf{u}/\partial t\) and \(\mathbf{A}_{t}\). It then substitutes the time derivative terms into the time Taylor expansion as:  
\[\overline{u}^{n+1}=\overline{\mathbf{u}}^{n}+\Delta t\overline{\mathbf{u}}_{t}^{n}+ \frac{\Delta t^{2}}{2}\overline{\mathbf{u}}_{tt}^{n}+\frac{\Delta t^{3}}{6} \overline{\mathbf{u}}_{ttt}^{n}. \tag{19}\]  
1.2 The Calculation of High-Order Temporal Derivatives by the Finite Difference Method Outside the Element  
Although the computational cost of the optimized TDG method is lower than that of the conventional TDG method, the computational complexity of the high-order temporal derivatives' estimation process remains significant. In Equations (33) and (34), the computation of spatial differentiation is necessary to estimate high-order temporal derivative terms \(\mathbf{U}_{tt}^{n}\) and \(\mathbf{U}_{ttt}^{n}\). To this end, this study uses the computational strategy optimized in [8], which applies the FDM to simplify the spatial differentiation calculation procedure.