# An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation
## 5 Results of Numerical Experimentation
### Comparison of the IHOFD Scheme under Asymmetric Boundary Conditions  
For comparison, take the square 2D model like the same one as before for example. An exact solution of the two-dimensional transient HCEs shown in Equation (1) in the conditions of Equation (21) was given in [26]. Here, the solution of the reference [26] is easily extended to satisfy the non-homogeneous boundary conditions. The asymmetric boundary conditions and the initial condition are  
\[ \begin{array}{l}T(0,y,t)=50\ ^{\circ}\text{C},T(1,y,t)=30\ ^{\circ}\text{C},\ t>0\\ T(x,0,t)=20\ ^{\circ}\text{C},T(x,1,t)=100\ ^{\circ}\text{C},\ t>0\\ T(x,y,0)=100\ ^{\circ}\text{C},\end{array}  \tag{21}\]  
Here, the thermal parameters and model parameters are the same as in the previous section. When the mesh size and the time step are the same as that of the fourth-order IHOFD algorithms in Section 5.3, the RE and MAE between the simulation result of the fourth-order IHOFD algorithm and the exact solution under the asymmetric boundary conditions are 6.082 \(\times\) 10\({}^{-4}\) and 1.412 \(\times\) 10\({}^{-2}\) (\({}^{\circ}\)C) in Figure 7. Compared with the result of the fourth-order IHOFD algorithm (see Table 6), it can be seen that the accuracy of the two is almost the same. It can be concluded that the IHOFD algorithm is also valid for the asymmetric boundary conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithms** & \begin{tabular}{c} **4 th-Order,** \\ **1.4\(\Delta\)t\({}_{1}\)** \\ \end{tabular} & \begin{tabular}{c} **4 th-Order,** \\ **2\(\Delta\)t\({}_{1}\)** \\ \end{tabular} & \
\begin{tabular}{c} **6 th-Order,** \\ **1.6\(\Delta\)t\({}_{2}\)** \\ \end{tabular} &
\begin{tabular}{c} **6 th-Order,** \\ **2\(\Delta\)t\({}_{2}\)** \\ \end{tabular} \\ \hline \(\text{RE}\times 10^{-4}\) & 4.240 & 6.063 & 4.275 & 5.348 \\ MAE (\({}^{\circ}\)C) \(\times 10^{-2}\) & 0.987 & 1.411 & 0.996 & 1.245 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The RE and MAE of the numerical simulation results of the IHOFD algorithm with different orders (4 th- and 6 th-order) and time steps and the exact solution of two-dimensional transient HCEs. To compare the results of the IHFOD algorithm for different grid densities (sizes), here, the thermal parameters are the same as in the above. the calculation domain was divided into 81 \(\times\) 81 rectangular mesh nodes with a mesh size of 0.0125 m, the time is 1.0 s, and the stability Equation (19b) determines the time step \(\Delta t\). The RE and MAE between the simulation result of the fourth-order IHOFD algorithm with a grid density of 81 \(\times\) 81 and the exact solution are 2.675 \(\times\) 10\({}^{-5}\) and 3.633 \(\times\) 10\({}^{-3}\) (\({}^{\circ}\)C) in Figure 8, respectively. The simulation errors of the fourth-order IHOFD algorithm with a grid density of 81 \(\times\) 81 are significantly smaller compared to that with a grid density of 41 \(\times\) 41 (see Figure 7). It can be concluded that the simulation accuracy of the IHOFD algorithm also improves significantly as the grid density increases. Figure 8: Numerical simulation of the 4 th-order IHOFD algorithm with a grid density of 81 \(\times\) 81 and the exact solution of two-dimensional HCEs (the centerline of \(y\) = 0.5 m is distributed along the \(x\) axis, with temperature in (\({}^{\circ}\)C)). Figure 7: Numerical simulation of the fourth-order IHOFD algorithm under the asymmetric boundary conditions and the exact solution of two-dimensional HCEs (the centerline of \(y\) = 0.5 m is distributed along the \(x\) axis, with temperature in (\({}^{\circ}\)C)). In summary, the main advantages of the proposed algorithms can be obtained as follows: (1) The stability factor of the LHOFD algorithm is slightly higher than that of the conventional algorithm (such as the GHOFD algorithm [26]). The stability factor of the IHOFD algorithm is twice the conventional algorithm, and the maximum absolute error of the thermal simulation is within 0.015 (\({}^{\circ}\)C).