## 2 Basic Theory
### The Comparison of the Conventional TDG and Optimized TDG  
In this section, a comparison is made between the conventional TDG and the optimized TDG (outside form and inside form) by expressing their solutions in an explicit form.  
For the conventional TDG method, substituting Equations (11) and (12) into Equation (6), the solutions can be expressed as:  
\[\begin{array}{ll}\overline{\mathbf{u}}^{n+1}=\overline{\mathbf{u}}^{n}+\Delta t& \left(\int_{x^{(k-1)}}^{x^{(k)}}\frac{\partial\mathbf{\phi}}{\partial x} \overline{\mathbf{f}}^{n}dx-\mathbf{\phi}\overline{\mathbf{f}}^{n}_{x}|_{x^{(k-1)}}^{x^{( k)}}dx\right)\\ &+\frac{1}{2}\Delta t^{2}\left(\mathbf{\Phi}\mathbf{A}^{n}\overline{\mathbf{n}}^{n}_{x}|_ {x^{(k-1)}}^{x^{(k)}}+\int_{x^{(k-1)}}^{x^{(k)}}\frac{\partial\mathbf{\phi}}{ \partial x}\mathbf{A}^{n}\overline{\mathbf{f}}^{n}_{x}dx\right)\\ &+\frac{1}{6}\Delta t^{3}\mathbf{\Phi}\left(\mathbf{A}^{n}\frac{\partial}{\partial x} \left(A^{k}_{n}\frac{\partial^{n}_{x}}{\partial x}\right)+\frac{\partial\mathbf{A }^{n}}{\partial\overline{\mathbf{f}}}\left(f^{n}_{x}\right)^{2}\right)|_{x^{(k-1) }}^{x^{(k)}}\\ &+\frac{1}{6}\Delta t^{3}\int_{x^{(k-1)}}^{x^{(k)}}\frac{\partial\mathbf{\phi}}{ \partial x}\left[A^{n}\frac{\partial}{\partial x}\left(\mathbf{A}f^{n}_{x}\right) +\frac{\partial\mathbf{A}}{\partial u}\left(f^{n}_{x}\right)^{2}\right]dx.\end{array} \tag{32}\]  
The above results clearly represent a complete integral form, obtained through spatial integration using the weak formulation. To achieve high-accuracy solutions, \(\partial\mathbf{A}\partial\mathbf{U}\) and repeated flux calculations were employed.  
For optimized TDG with outside finite difference method, substituting Equations (17), (22), and (23) into Equation (19), the solutions can be expressed as:  
\[\begin{array}{ll}\overline{\mathbf{u}}^{n+1}_{(k,i)}=\overline{\mathbf{u}}^{n}_{(k, i)}+\Delta&t\int_{x^{(k-1)}}^{x^{(k)}}\frac{\partial\mathbf{\Phi}}{\partial x} \overline{\mathbf{f}}^{n}_{(k,i)}dx-\mathbf{\Phi}\overline{\mathbf{f}}^{n}_{(k,i)}dx\\ &-\frac{\Delta t^{2}}{2}\frac{1}{12\Delta x}\left(G^{n}_{(k-2,i)}-8G^{n}_{(k-1) }+8G^{n}_{(k+1,i)}-G^{n}_{(k+2,i)}\right)\\ &-\frac{\Delta t^{3}}{6}\frac{1}{12\Delta x}\left(H^{n}_{(k-2,i)}-8H^{n}_{(k- 1,i)}+8H^{n}_{(k+1,i)}-H^{n}_{(k+2,i)}\right).\end{array} \tag{33}\]  
For optimized TDG with the inside finite difference method, substituting Equations (17) and (31) into Equation (19), the solutions can be expressed as:  
\[\overline{\mathbf{u}}^{n+1}_{(i)}=\overline{\mathbf{u}}^{n}_{(i)}+\Delta t\int_{x^{(k -1)}}^{x^{(k)}}\frac{\partial\mathbf{\Phi}}{\partial x}\overline{\mathbf{f}}^{n}_{(i )}dx-\mathbf{\Phi}\overline{\mathbf{f}}^{n}_{(i)}\frac{\partial\overline{\mathbf{f}}^{n}_ {(i)}}{\partial x}\Bigg{|}_{x^{(k-1)}}^{x^{(k)}}+\frac{\Delta t^{2}}{2}(\mathbf{a }\mathbf{G}^{n})+\frac{\Delta t^{3}}{6}\mathbf{a}\mathbf{H}^{n}. \tag{34}\]  
Compared to Equation (32), Equations (33) and (34) do not utilize \(\partial\mathbf{A}/\partial U\) but instead employ the simpler operator \(\partial\mathbf{A}/\partial t\), as explained in Section 2.1.2. Furthermore, from Equation (33), it can be observed that the optimized TDG method is a hybrid method combining the DG method and the FDM method. During the application of the FDM method, Equation (33) uses information from elements surrounding the target element for the finite difference calculations. In contrast, Equation (34) demonstrates a way of performing differential calculations using information inside the target element. Clearly, the approach of estimating high-order time derivative terms using internal element information provides better compactness.