# An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation
## Appendix B The Formula for the HONFD Coefficients of the Fourth-Order Spatial Derivative  
An HONFD operator of the fourth-order derivative in space with the \((M-3)\)-order-accuracy can be approximated as  
\[\frac{\partial^{4}f(x)}{\partial x^{4}}\approx\frac{1}{\Delta x^{4}}\sum_{i=1 }^{M}d_{i}f(x_{0}+m_{i}\Delta x)\] (A6)  
The general form of the HONFD operator can be also expressed as a vector \(g=[g_{1},g_{2},\cdots,g_{N}]\), \(M\) represents the length of the operator, and \(M\) is an odd number greater than or equal to 5. Where \(g_{ii}=m_{ii}\), \(m=[-L_{p},-L_{p}+1,\cdots,0,1,2,\cdots,L_{q}]\) and \(L_{p}+L_{q}=M-1\), \(L_{p}\), \(L_{q}\) are all positive integers, and \(m\) is also a vector. \(ii\) represents the ordinal numbers of the vector elements of the HONFD operator from left to right, \(ii=1,2,\cdots,M\), and \(\Delta x\) represents the mesh size along the \(x\) coordinate.  
Similar to the above, we obtain the following expression as  
\[\begin{array}{l}\frac{\partial^{4}f(x_{0})}{\partial x^{4}}=\frac{1}{\Delta x ^{4}}[\sum\limits_{ii=1}^{M}\sum\limits_{i=1}^{M-3}\frac{(m_{ii})^{i}\Delta x^ {i}}{(i)!}d_{ii}f^{(i)}(x_{0})\\ +\sum\limits_{ii=1}^{M}\frac{(m_{ii})^{M-2}}{(M-2)!}d_{ii}\Delta x^{M-2}f^{(M-2 )}(x_{0})+O(\Delta x^{M-1})]\end{array}\] (A7)  
where \(m\) represents the number of mesh points, \(m=0\) indicates the derivation at the location \(m=0\). \(f^{(i)}(x_{0})\) represents \(f(x_{0})\) with a derivative of the order \(i\) at the location \(x_{0}\).  
Equation (A7) can be expressed as a matrix form and simplified to obtain  
\[\begin{array}{l}\begin{pmatrix}1&g_{1}&g_{1}^{2}&\cdots&g_{1}^{M-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 1&g_{5}&g_{5}^{2}&\cdots&g_{5}^{M-1}\\ 1&g_{6}&g_{6}^{2}&\cdots&g_{6}^{M-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 1&g_{M}&g_{M}^{2}&\cdots&g_{M}^{M-1}\end{pmatrix}\begin{pmatrix}d_{1}\\ \vdots\\ d_{5}\\ d_{6}\\ \vdots\\ d_{M}\end{pmatrix}=\begin{pmatrix}0\\ \vdots\\ 4!\\ 0\\ \vdots\\ 0\end{pmatrix}\end{array}\] (A8)  
The coefficient matrix \(V\) is a \(M\times M\) Vandermonde matrix. Directly using [PERSON]'s rule, Equation (A8) can be easily solved to obtain the HONFD coefficients \(d_{ii}\). However, for a higher-order Vandermonde problem, it is quite commonly a singularity, thus affecting the accuracy of the HONFD coefficients. Inspired by [32], we develop a general HONFD coefficient formula in different ways, so as to solve the problem well.  
Consider Equation (A8) as a polynomial function, that is  
\[\varphi(z)=\sum_{i=1}^{M}d_{i}z^{i-1}\] (A9)  
and \(\varphi(g_{i})=d_{i}\), \(i=1,2,\cdots,M\), \(d_{i}\) denotes the column vector or column matrix element, where \(d_{5}=4!\), and the other is zero. The elements of the vector of the HONFD operator \(g_{1}\),\(g_{2}\),\(\cdots,g_{M}\) are different from each other. Thus, the coefficient matrix determinant \(|V|\
eq 0\), therefore \(f(z)\) can be expressed as the Lagrange interpolation formula, i.e.,\[\varphi(z)=\sum\limits_{i=1}^{M}d_{i}\frac{\prod\limits_{\begin{subarray}{c}k=1 \times\
eq i\\ k=1,k\
eq i\end{subarray}}^{M}(z-g_{k})}{\prod\limits_{\begin{subarray}{c}k=1,k\
eq i\end{subarray}}^{M}(g_{i}-g_{k})} \tag{10}\]  
Expanding the numerator of Equation (10) into a polynomial form and comparing it with Equation (9), we obtain  
\[d_{i}=4!\left(\frac{r_{M-5}\cdots-r_{1}\geq 1\leq r_{1},\cdots,r_{M-5}\
eq i}{ \prod\limits_{\begin{subarray}{c}k=1,k\
eq i\end{subarray}}^{M}(g_{k}-g_{i})}\right) \tag{11}\]  
where \(r_{1},r_{2}\cdots,r_{M-5}\) represents different natural numbers.  
From Equation (A7), we can obtain the truncation error coefficient of the HONFD operator of the fourth-order derivative in space with \((M-3)\)-order-accuracy, namely  
\[e_{M}=|\frac{1}{(M-2)!}\sum\limits_{i=1}^{M}{(m_{i})}^{M-2}d_{i}| \tag{12}\]