# An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation
## 3 Schemes Analysis
### Accuracy Analysis  
In the wavenumber domain, the \(2M\)-order HOCFD operator of a one-variable function (see Appendix A (A1)) can be expressed as  
\[kh\approx(4\sum_{i=1}^{M+1}b_{i}\ \sin^{2}(\frac{kih}{2}))^{1/4} \tag{14}\]  
Among them, \(h\) represents the size of the spatial mesh, \(k\) indicates the wavenumber, and \(b_{i}\) indicates the HOCFD coefficient. Let  
\[f(kh)=(4\sum_{i=1}^{M+1}b_{i}\ \sin^{2}(\frac{kih}{2}))^{1/4} \tag{15}\]  
According to [PERSON]'s law of sampling, there is \(k_{\text{max}}=\pi/h\). According to the HOCFD coefficients \(b_{i}\) of different orders, Equation (15) can be used to obtain the comparison relationship between the accuracy of the different-order HOCFD operators and exact solution (see Figure 1). It can be concluded that the accuracy of the HOCFD operator can be significantly improved with the increase in the different-order number. The HOFD operator of the same order of the fourth-order spatial derivatives are less accurate than the second-order spatial derivatives. Figure 1: Comparison chart of the accuracy of the 4 th–20 th-order HOCFD operators of the different order spatial derivative and exact solution: (**a**) 2 nd-order spatial derivative [(26)] and (**b**) 4 th-order spatial derivative. The qualitative evaluation is previously given. Based on Equation (16a) [26] and Equation (16b), the \(\left(kh\right)_{\text{max}}\) values corresponding to different-order HOCFD operators of the second- and fourth-order spatial derivative with errors less than or equal to \(\varepsilon\) can be solved, respectively. According to these values, the accuracy of different-order HOCFD operators can be quantitatively evaluated. \[\min\lvert kh-2(\sum\limits_{i=1}^{M}a_{i}\ \sin^{2}(\frac{kih}{2}))^{1/2}\rvert<\varepsilon\] (16a) and \[\min\lvert kh-(4\sum\limits_{i=1}^{M+1}b_{i}\ \sin^{2}(\frac{kih}{2}))^{1/4}\rvert<\varepsilon \tag{16b}\]  
From Tables 2 and 3, with the admissible error reduction, the \(\left(kh\right)_{\text{max}}\) value corresponding to the different HOCFD operator of the same order significantly decreases, and under the same admissible error conditions, the \(\left(kh\right)_{\text{max}}\) value corresponding to the HOCFD operator of the second-order derivative in space under the same order is a little greater than the value corresponding to the fourth-order spatial derivative; in other words, under the same order, the accuracy of the HOCFD operator of the second-order spatial derivative is higher than that of the fourth-order spatial derivative. In addition, based on the truncation error coefficient Formula (17) of the 2M-order HOCFD operator of the fourth-order spatial derivation (see Appendix A for detailed derivation),  
\[e_{M}=\lvert\frac{2}{(2M+4)!}\sum\limits_{i=1}^{M+1}i^{2M+4}b_{i}\rvert \tag{17}\]  
The curve of the truncation error (TE) coefficient of the different-order HOCFD operators can be obtained (see Figure 2). The TE coefficient of the different-order HOCFD operator swiftly decreases with an increasing order, as shown in Figure 2; in other words, its accuracy rapidly increases. When 2\(M\) = 10, the TE coefficients of the 10 th-order HOCFD operators of the 2 nd-order and the 4 th-order spatial derivations are 6.013 \(\times\) 10\({}^{-5}\) (\(e_{M}=\lvert\frac{2}{(2M+2)}\sum\limits_{i=1}^{M}i^{2M+2}a_{i}\rvert\)[26]) and 2.128 \(\times\) 10\({}^{-4}\) (see the arrow in Figure 2), respectively. It can be concluded that the accuracy of the HOCFD operator can be significantly improved with the increase in the different-order number. And, under the same order, the TE coefficient of the HOCFD operator of the fourth-order spatial derivative is significantly higher than that of the second-order spatial derivative.