# An Efficient Finite-Difference Stencil with High-Order Temporal Accuracy for Scalar Wave Modeling
## 6 Conclusions  
In this work, we proposed a new FD stencil with high-order temporal and spatial accuracies. The new stencil can reach at least sixth-order temporal accuracy and involves fewer grid nodes than that of the well-known cross-rhombus stencil. We presented two methods to solve the FD coefficients of the new stencil by Taylor-series expansion and the dispersion relationship-preserving method, in which the DRP-based FD coefficients can effectively mitigate the dispersion errors from the high-wavenumber components. Dispersion, stability analyses and numerical experiments demonstrated that the new stencil had better accuracy and stability than those of the conventional cross-stencil. Numerical accuracy and execution time analyses proved that the accuracy of the new stencil was slightly lower than that of the standard cross-rhombus stencil in the high-order cases, but its computational efficiency was greatly improved. Thus, the new stencil is a more economical and feasible method for large-scale modeling.  
Conceptualization, G.C. and Y.L.; methodology, G.C.; validation, Z.P.; formal analysis, G.C.; investigation, G.C.; resources, Y.L.; writing--original draft preparation, G.C.; writing--review and editing, Z.P. and Y.L.; visualization, G.C.; supervision, Z.P.; project administration, Z.P.; funding acquisition, Z.P. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Natural Science Foundation of the Sichuan Province of China (Grant No. 2022 NSFSC40574) and partially supported by the National Natural Science Foundation of China (Grant No. 61775030 and Grant No. 61571096).  
Figure 8: Seismogram at position \((0,2125)\) (m) of different FD methods. Trace **a** represents the record of the TE-C method. Trace **b** represents the TE-R method. Trace **c** represents the DRP-R method. Trace **r** represents the reference trace. Trace **a-r** represents the relative error between trace **a** and reference trace **r**. Trace **b-r** represents the relative error of trace **b**. Trace **c-r** represents the relative error of trace **c**.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Orders** & \multicolumn{4}{c}{**Total Relative Errors (Pa)**} \\ \hline M & N & TE-C & TE-CR & TE-R & DRP-R \\ \hline
8 & 8 & 7.974763 & 2.203247 & 2.348245 & 0.715832 \\
8 & 4 & \(\backslash\) & 2.390345 & 2.499854 & 0.846557 \\
6 & 6 & 12.522196 & 4.380239 & 4.473768 & 1.091585 \\
6 & 3 & \(\backslash\) & 4.502392 & 4.628571 & 1.154466 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Relative errors of different FD methods in the 2D Sigsbee velocity model.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** This study benefited from the reproducible codes and documentation provided in Madagascar open-source software [27]. The authors appreciate the owner of the copyrights.  
**Conflicts of Interest:** The authors declare no conflict of interest.The funders had no role in the design of the study or in the collection or analyses.