Improvement of high-order spatiotemporal finite difference method for three-dimensional acoustic wave equations
Authors: Yi Tianyu , Luan Xiwu , Meng Fanshun , Fang Gang 
Published: January 2025

Abstract
The finite-difference (FD) method has become one of the most widely used numerical simulation techniques in the field of seismic exploration due to its simplicity, high computational efficiency, and low storage requirements. Generally, the numerical approximation of continuous differential operators in wave equations using discrete difference operators can lead to numerical dispersion in both time and space domain. While increasing the length of the spatial operator can facilitate high-order accuracy in spatial numerical simulations, a phenomenon known as the “saturation effect”, that the rate of improvement in spatial approximation accuracy diminishes as the operator length increases, and excessive computational effort does not necessarily enhance the overall accuracy of the numerical simulation. Improving the accuracy of spatial approximation alone cannot meet the requirements for numerical simulation accuracy. Additionally, enhancing the accuracy of time approximation is quite challenging, as it not only significantly increases computational complexity but also greatly reduces the stability of numerical simulations. It has become a significant research focus in FD numerical simulation to enhance the accuracy of time-domain simulations while maintaining acceptable computational complexity and ensuring that the stability conditions of the numerical methods are easily satisfied. To address the challenges, we propose an improved high-precision FD method grounded in the dispersion relationship within the spatiotemporal domain. This method utilizes a FD octahedral stencil derived from the scalar acoustic wave equation (AWE). We enhance the original octahedral stencil by expanding the two independent operators to three independent operators, while constraining the length of the third operator to its minimum value. This adjustment reduces the number of difference coefficients that need to be calculated, significantly improving the efficiency of numerical simulations. Our findings demonstrate that the reduction in operator length does not compromise accuracy and enhances the stability of the algorithm, as evidenced by analyses of numerical dispersion curves and numerical simulation experiments. We derive a difference scheme for a three-dimensional mixed absorption boundary, and the numerical results demonstrate that the artificial boundary exhibits effective absorption. Furthermore, to enhance efficiency and ensure accuracy, a new stencil is integrated with the implicit difference method in the spatial domain, resulting in a novel FD stencil that combines spatial implicit differences with time explicit differences in the spatiotemporal domain. Analysis of the numerical dispersion curves and numerical simulation results indicate that this new method improves numerical dispersion accuracy in the spatiotemporal domain while maintaining efficiency. The new implicit FD stencil employs a recursive filter for the implicit difference explicit solution. As a result, under the same accuracy conditions, the new implicit FD method requires the least operational time, utilizes the minimum spatial operator length, and achieves the highest simulation accuracy, thereby providing a dual improvement in both accuracy and efficiency.