# An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation
## 5 Results of Numerical Experimentation
### Comparison of the Different-Order LHOFD Schemes  
Assuming the thermal conductivity \(\alpha=1\) m\({}^{2}\)-s\({}^{-1}\), the calculation domain was divided into 41 \(\times\) 41 rectangular mesh nodes with a mesh size of 0.025 m, time of 1.0 s, and stability factors for the fourth-order and sixth-order LHOFD algorithms of 0.1938 and 0.1713, respectively, corresponding to the upper limit of the time steps determined by the stability Equation (18c), which are 1.211 \(\times\) 10\({}^{-4}\) s and 1.071 \(\times\) 10\({}^{-4}\) s, respectively. It can be concluded that the temperature distribution curves using a numerical simulation of the LHOFD algorithm with different orders (fourth and sixth order) coincide with that of the exact solution along the \(x\) axis of the center line \(y\) = 0.5 m in Figure 6. The relative error (RE) calculated by \(E=\big{(}\sum\limits_{i=1}^{T_{M}}\left(T_{exact}-T_{cal}\right)^{2}\big{/} \sum\limits_{i=1}^{T_{M}}T_{exact}^{2}\big{)}^{1/2}\), where \(T_{M}\) represents the total point numbers, \(T_{cal}\) represents the numerical simulation results, and \(T_{exact}\) represents the exact solution, and the maximum absolute error (MAE) determined by the error maximum norm of the numerical simulation results of the LHOFD algorithm with different orders (fourth and sixth order) and the exact solution of two-dimensional transient HCEs are 2.702 \(\times\) 10\({}^{-4}\), 6.281 \(\times\) 10\({}^{-3}\) (\({}^{\circ}\)C) and 2.156 \(\times\) 10\({}^{-4}\), 4.763 \(\times\) 10\({}^{-3}\) (\({}^{\circ}\)C), respectively.  
Figure 5: Numerical simulation results of two-dimensional HCEs by the LHOFD and IHOFD algorithms with different stability factors: (**a**) LHOFD algorithm with stability factor (\(F\) = 1.713 \(\times\) 10\({}^{-1}\)), (**b**) LHOFD algorithm with stability factor (\(F\) = 1.713 \(\times\) 10\({}^{-1}\) + 0.365 \(\times\) 10\({}^{-2}\)), (**c**) IHOFD algorithm with stability factor (\(F\) = 3.309 \(\times\) 10\({}^{-1}\)) and (**d**) IHOFD algorithm with stability factor (\(F\) = 3.309 \(\times\) 10\({}^{-1}\) + 0.017 \(\times\) 10\({}^{-1}\)).  
When the time step is 1.071 \(\times\) 10\({}^{-4}\) s, the RE and MAE of the numerical simulation results of the fourth-order LHOFD algorithm and the exact solution of two-dimensional transient HCEs are 2.535 \(\times\) 10\({}^{-4}\), 5.569 \(\times\) 10\({}^{-3}\) (\({}^{\circ}\)C), respectively. It can be seen from the above that the LHOFD algorithm is correct and effective, and with the increase in the order of the LHOFD algorithm, its simulation accuracy is significantly improved. At the same time, the computational complexity and cost of the LHOFD algorithm also increases significantly.  
In addition, under the same conditions, the RE and MAE of the numerical simulation results of the fourth-order GHOFD algorithm (see [26]) are 7.525 \(\times\) 10\({}^{-7}\), 3.606 \(\times\) 10\({}^{-5}\) (\({}^{\circ}\)C), respectively, which are much smaller than those of the LHOFD algorithm results. These results are consistent with the theoretical analysis results presented in Section 3.1.