# An Efficient Finite-Difference Stencil with High-Order Temporal Accuracy for Scalar Wave Modeling
## Appendix A  
In this part, we analyze the accuracy of our new stencil with the time-space domain. TE-based FD coefficients (Equation (12)). First, we define the absolute error \(\varepsilon\) according to Equation (6) as follows:  
\[\varepsilon=\left|\begin{array}{l}a_{0}+2\sum_{m=1}^{M}a_{m}[\cos(mk_{x}h)+ \cos(mk_{z}h)]\\ +4\sum_{n=1}^{N-1}b_{n}[\cos(nk_{x}h)\cos(nk_{z}h)]-r^{-2}[-2+2\cos(\omega \Delta t)]\end{array}\right|. \tag{13}\]  
By substituting Equations (7) and (10) into Equation (13), we obtain  
\[\varepsilon =\left|\begin{array}{l}a_{0}+2\sum_{m=1}^{M}\left[a_{m}\sum_{j= 0}^{\infty}\frac{(-1)^{j}m^{2}(k_{x}^{2}+k_{z}^{2})h^{2j}}{(2j)!}\right]+4 \sum_{n=1}^{N-1}\left[b_{n}\sum_{j=1}^{\infty}\frac{(-1)^{j}n^{2}(k_{x}^{2}+k_ {z}^{2})h^{2j}}{(2j)!}\right]\\ +4\sum_{n=1}^{N-1}\left[b_{n}\sum_{\xi=1}^{\infty}\sum_{\xi=1}^{\infty}\frac{ (-1)^{j+\xi}n^{2}(\xi+\xi)}{(2\xi)!(2\xi)!}k_{x}^{2}k_{x}^{2}h^{2}(\xi+\xi) \right]-2r^{-2}\sum_{j=1}^{\infty}\frac{(-1)^{j}r^{2j}(kh)^{2j}}{(2j)!}\end{array}\right|\] \[=\left|\begin{array}{l}a_{0}+4\sum_{m=1}^{M}a_{m}+4\sum_{n=1}^{N -1}b_{n}\\ +2\sum_{j=1}^{\infty}(-1)^{j}\left[\sum_{m=1}^{M}a_{m}\frac{m^{2}(k_{x}^{2}+ k_{z}^{2})}{(2j)!}+2\sum_{n=1}^{N-1}b_{n}\frac{n^{2j}(k_{x}^{2}+k_{z}^{2})}{(2j)!} \right]h^{2j}\\ +4\sum_{n=1}^{N-1}\left[b_{n}\sum_{\xi=1}^{\infty}\sum_{\xi=1}^{\infty}\frac {(-1)^{j+\xi}n^{2}(\xi+\xi)}{(2\xi)!(2\xi)!}k_{x}^{2}k_{x}^{2}h^{2}(\xi+\xi) \right]\\ +4\sum_{n=1}^{N-1}\left[b_{n}\sum_{\xi=1}^{\infty}\sum_{\xi=1}^{\infty}\frac {(-1)^{j+\xi}n^{2}(\xi+\xi)}{(2\xi)!(2\xi)!}k_{x}^{2}k_{x}^{2}h^{2}(\xi+\xi) \right]-2r^{-2}\sum_{j=1}^{\infty}\frac{(-1)^{j}r^{2j}(kh)^{2j}}{(2j)!}\end{array}\right|\] \[=\left|\begin{array}{l}a_{0}+4\sum_{m=1}^{M}a_{m}+4\sum_{n=1}^{ N-1}b_{n}\\ +2\sum_{j=1}^{\infty}(-1)^{j}\left[\sum_{m=1}^{M}a_{m}\frac{m^{2j}(k_{x}^{2}+k_{z}^{2})}{(2j)!}+ 2\sum_{n=1}^{N-1}b_{n}\frac{n^{2j}(k_{x}^{2}+k_{z}^{2})}{(2j)!}\right]h^{2j}\\ +4\sum_{n=1}^{N-1}\left[b_{n}\sum_{\xi=1}^{\infty}\sum_{\xi=1}^{\infty}\frac {(-1)^{j+\xi}n^{2}(\xi+\xi)}{(2\xi)!(2\xi)!}k_{x}^{2}k_{x}^{2}h^{2}(\xi+\xi) \right]\\ -2r^{-2}\sum_{j=1}^{\infty}\frac{(-1)^{j}(k_{x}^{2}+k_{z}^{2})h^{2j}}{(2j)!}-2 r^{-2}\sum_{j=1}^{\infty}\sum_{\xi=1}^{j-1}\frac{(-1)^{j}r^{2j}k_{x}^{2}k_{x}^{2j -2}j!h^{2j}}{(2j)!(j-\xi)!}\end{array}\right| \tag{14}\] \[\approx\left|\begin{array}{l}2\sum_{j=M+1}^{\infty}(-1)^{j} \left[\sum_{m=1}^{M}a_{m}\frac{m^{2j}(k_{x}^{2}+k_{z}^{2})}{(2j)!}+2\sum_{n=1} ^{N-1}b_{n}\frac{n^{2j}(k_{x}^{2}+k_{z}^{2})}{(2j)!}\right]h^{2j}\\ +4\sum_{n=1}^{N-1}\left[b_{n}\sum_{\xi=1,\xi=1,\xi+\xi>3}\frac{(-1)^{j+\xi}n^{ 2(\xi+\xi)}}{(2\xi)!(2\xi)!(2\xi)!}k_{x}^{2}k_{x}^{2}h^{2}(\xi+\xi)\right]\\ -2r^{-2}\sum_{j=M+1}^{\infty}\frac{(-1)^{j}2^{j}\Delta t^{2}(k_{x}^{2}+k_{z}^{2} )}{(2j)!}-2r^{-2}\sum_{j=4}^{\infty}\sum_{\xi=1,\xi+\mu int(j/2)}\frac{(-1)^{j} \omega^{2}\Delta t^{2}k_{x}^{2}k_{x}^{2}l^{2-2\xi}j!}{(2j)!(j)!(j-\xi)!}\end{array}\right|\] \[\approx\left[o(h^{2M})+o(h^{6})-o(\Delta t^{2M})-o(\Delta t^{6}) \right]\approx o(h^{6})+o(\Delta t^{6}).\]  
where the minimum power of \(h\) and \(\Delta t\) is six, and thus the modeling accuracy is of the sixth order.