# An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation
## 4 Integration Procedure
### Boundary Process for the High-Order FD Schemes  
When numerically solving HCEs utilizing an HOFD algorithm, it is important to calculate the derivative at the boundary point and its neighbor points [32]. The algorithms proposed in this paper include the HOCFD and HONFD methods of three different spatial derivatives, such as the second derivative, fourth derivative, and fourth mixed derivative. The HOCFD and HONFD methods of the second-order spatial derivative in the boundary zone are shown in reference [26], while the HOCFD and HONFD methods of the fourth-order spatial derivative are similar to that of the second-order spatial derivative. The boundary processing strategy of the fourth-order spatial mixed derivatives is highlighted here. For the Dirichlet boundary conditions, we present a new strategy of combination with the HOCFD and the HONFD approximations under the same high-order accuracy as the internal computational domain to calculate the derivatives of the boundary domains, which has very high accuracy and does not add an additional computational cost. Generally, the HOFD numerical calculation of the transient thermal field in a two-dimensional model includes two parts, namely the internal computational domain and the boundaries domain. A sketch of the internal computational domain \(\Omega_{0}\) and the boundaries domain \(\Omega_{k}\) of the two-dimensional model for the HOFD numerical simulation of the transient thermal field in two dimensions is displayed in Figure 3. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**2M Order** & **4** & **6** & **8** & **10** & **12** & **14** & **16** & **18** & **20** \\ \hline GHOFD’s & 1.875 & 1.654 & 1.538 & 1.465 & 1.414 & 1.376 & 1.346 & 1.323 & 1.303 \\ \hline LHOFD’s & 1.938 & 1.713 & 1.588 & 1.508 & 1.451 & 1.409 & 1.376 & 1.350 & 1.328 \\ \hline CHOFD’s & 3.750 & 3.309 & 3.076 & 2.930 & 2.828 & 2.752 & 2.693 & 2.645 & 2.606 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Stability factors (\(\times 10^{-1}\)) of the difference algorithms for two-dimensional transient HCEs. The width of the \(M-1\), the grid points, and the boundary domains are \(\Omega_{k}(k=1,2,\cdots,8)\), where \(M\) is half the length of the HOFD operator in the internal calculation domain. When the meshes of the two-dimensional model are \(x_{N}\times y_{N}\), \(x_{N},y_{N}\) represents the number of meshes along the \(x\) and \(y\) coordinates, respectively, and the internal computational domain \(\Omega_{0}:Lh\leq x\leq(x_{N}-2M)h,Lh\leq y\leq(y_{N}-2M)h\). \(h\) represents the mesh size in space. The HONFD operators in the boundary domains have the same order as the HOCFD operators in the internal computing domain. In the sixth-order LHOFD and IHOFD algorithms, the corner domain \(\Omega_{5}\) (see Figure 3) is taken as an example to briefly introduce the calculation method and the process of the fourth-order spatial mixed derivative in the boundary region. Specifically, the first step is to calculate the coefficients of HONFD along the \(x\) and \(y\) coordinates of four mesh points. The first point, marked as 1 in the corner domain, its corresponding stencil, and the sixth-order HONFD coefficients of the second-order spatial derivative along the \(x\) coordinate, are shown in Table 5 serial number 1, and the coefficients along the \(x\) coordinate are the same as those along the \(x\) coordinate. The second point marked as 2, its stencil, and the sixth-order HONFD coefficients in the \(x\) coordinate are shown in Table 5 by serial number 2, and the coefficients along the \(y\) coordinate are the same as those along the \(x\) coordinate. The third point marked as 3, the stencil, and the sixth-order HONFD coefficients along the \(x\) coordinate are the same as those of the second point, the stencil, and the sixth-order HONFD coefficients along the \(y\) coordinate are the same as those of the first point.