# An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation
## 5 Results of Numerical Experimentation
### Comparison of the Different Schemes  
First, the numerical simulation results of the IHOFD method with different orders under the same grid size condition are given, and their results are compared with the LHOFD method in this section. Secondly, the results obtained by the IHOFD method with different orders and time steps are given and discussed. 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- and sixth-order LHOFD algorithms (\(\Delta\)t\({}_{1}\) = 1.211 \(\times\) 10\({}^{-4}\) s, \(\Delta\)t\({}_{2}\) = 1.071 \(\times\) 10\({}^{-4}\) s), the RE and MAE between the results of the fourth- and sixth-order IHOFD algorithms and the exact solution are 3.126 \(\times\) 10\({}^{-4}\), 7.279 \(\times\) 10\({}^{-3}\) (\({}^{\circ}\)C) and 2.761 \(\times\) 10\({}^{-4}\), 6.435 \(\times\) 10\({}^{-3}\) (\({}^{\circ}\)C), respectively. Compared with the LHOFD algorithm (see Section 5.2), the simulation accuracy of the IHOFD algorithm of the same order is slightly lower under the same grid size and the same time step. This is mainly due to the fact that the IHOFD algorithm (see Equation (4)) omits other fourth-order spatial derivatives in the LHOFD algorithm (see Equation (2)) except for the fourth-order spatial mixed derivative term. When the grid size is the same, under different time steps (\(\Delta\)t\({}_{1}\) = 1.172 \(\times\) 10\({}^{-4}\) s, \(\Delta\)t\({}_{2}\) = 1.034 \(\times\) 10\({}^{-4}\) s), the errors analysis between the results of the fourth-order and sixth-order IHOFD algorithms and the exact solution are shown in Table 6. When the time step (1.4\(\Delta\)t\({}_{1}\) = 1.641 \(\times\) 10\({}^{-4}\) s) of the fourth-order IHOFD algorithm is slightly less than the time step (1.6\(\Delta\)t\({}_{2}\) = 1.654 \(\times\) 10\({}^{-4}\) s) of the sixth-order, the accuracy of the former is still slightly higher than that of the latter. And, when the time step (2\(\Delta\)t\({}_{1}\)) of the fourth-order IHOFD  
Figure 6: Numerical simulation comparison of the LHOFD method with different orders (fourth and sixth) and the exact solution of two-dimensional HCEs (the centerline of \(y\) = 0.5 m is distributed along the \(x\) axis, temperature in (\({}^{\circ}\)C)). algorithm is greater than the time step (2\(\Delta\)t\({}_{2}\)) of the sixth-order, the latter has a slightly higher accuracy compared to the former. When the time step reaches the upper limits (2\(\Delta\)t\({}_{1}\), 2\(\Delta\)t\({}_{2}\)) of the stability condition, the simulation accuracy of the fourth-order and sixth-order IHOFD algorithms decreases to a certain extent, but their accuracy errors are less than \(\pm\)0.015 (\({}^{\circ}\)C). When the mesh size and time step (\(\Delta\)t = 2 \(\times\) 10\({}^{-4}\) s) remain the same, the REs and MAEs between the fourth-order and sixth-order IHOFD algorithm and the exact solution are 5.172 \(\times\) 10\({}^{-4}\), 1.2036 \(\times\) 10\({}^{-2}\) (\({}^{\circ}\)C) and 5.171 \(\times\) 10\({}^{-4}\), 1.2040 \(\times\) 10\({}^{-2}\) (\({}^{\circ}\)C), respectively. It can be seen that the accuracy of the two is almost the same. It can be concluded from above that the simulation accuracy of the fourth-order and sixth-order IHOFD algorithms is very close and the simulation accuracy improvement of the latter is very small when the grid size is the same. Based on the comprehensive analysis of stability (see Table 4), accuracy, and computational cost, it can be concluded that the fourth-order IHOFD algorithm is more advantageous than the sixth-order IHOFD algorithm.