# An Efficient Finite-Difference Stencil with High-Order Temporal Accuracy for Scalar Wave Modeling
## 1 Introduction  
Finite-difference methods are the most popular tool for understanding complex wave phenomena due to their straightforward implementation and relatively small computational cost [1; 2; 3; 4; 5; 6; 7]. Solving scalar wave equations through FD methods is a basic step in seismic depth migration [8] and velocity modeling [9; 10; 11]. An FD stencil defines the grid nodes related to the FD operator, such as the widely used cross-stencil involving a series of horizontal and vertical grid nodes. High-order FD stencils for approximating temporal derivatives have practical advantages in mitigating dispersion errors and stability problems.  
To obtain an FD stencil with high-order time approximation, the authors of [1] presented a Lax-Wendroff scheme in which the high-order FD operators of the temporal derivatives are transformed into the FD operators of the mixed spatial derivatives. The authors of [12; 13] further developed this method with fourth- and sixth-order temporal accuracies and derived a stability formula for the scalar wave equation. Afterward, in [14], the authors presented a new FD stencil with a rhombus shape. The rhombus stencil is also similar to the Lax-Wendroff method, and it can reach arbitrary even-order accuracy for both the temporal and spatial accuracies. To improve the computational efficiency, the authors of [15] presented a mixed stencil with a small rhombus stencil and a large cross stencil. They demonstrated that this cross-rhombus stencil can reach (2M)th-order spatial and (2N)th-order temporal accuracies with the time-space domain FD coefficients. Theauthors of [16; 17] further developed the cross-rhombus stencil in a staggered grid and gave the analytical formula for solving FD coefficients with fourth- and sixth-order temporal accuracies. Then, the authors of [18] presented a method for solving FD coefficients with arbitrary even-order accuracies in a staggered grid and investigated the optimal FD coefficients using a combination of the Taylor series expansion and least squares method.  
A standard cross-rhombus stencil with a high-order time approximation will involve more grid nodes, thus increasing the cost exponentially. For the problem of low computational efficiency in high-order cases, we propose a new temporal high-order FD stencil. The new stencil has a radial shape, including a standard cross-stencil and a rotated cross-stencil with a \((\pi/4)\) degree, in which the rotated cross stencil is similar to the Lax-Wendroff approach to improve the temporal accuracy. Compared with the standard cross-rhombus stencil method, the new stencil involves fewer grid nodes, and this is significant for reducing computational costs in large-scale seismic modeling. We prove that the new stencil can reach sixth-order temporal accuracy with the time-space domain FD coefficients. Although the new stencil cannot guarantee higher accuracy than the sixth-order time approximation, it is a compromise scheme which sacrifices a little accuracy in exchange for a large increase in computational efficiency. Then, we use the Taylor-series expansion (TE) method [19] to obtain the FD coefficients of the proposed FD stencil. As an alternative, we present the dispersion relationship-preserving (DRP) method [6; 7; 20; 21] to obtain the FD coefficients satisfying more elevated wavenumbers in the dispersion relationship. The DRP-based FD coefficients can effectively mitigate the dispersion errors from the high-wavenumber components. To verify the feasibility of the proposed method, we designed a series of experiments to analyze the performance of the new stencil. The results indicate that the new stencil is a significant improvement in mitigating dispersion error and computational efficiency compared with the conventional methods.