三维波传播模拟的多项式特解法及其物理应用

Polynomial Special Solution Method for Three-Dimensional Wave Propagation Simulation and Its Physical Applications

摘要: 本文提出一种基于多项式特解的三维波动方程高精度数值方法，探究在不同传播速率下三维波动方程的数值模拟问题。该方法使用满足波动方程的不同阶的多项式特解作为基函数对波动方程进行数值模拟，本文选取传播速率存在差异的数值算例展开求解并与有限差分法进行对比，结果表明：该算法比传统有限差分法精度更高，易于实现，两个算法的计算误差随波速减小(对应物理波长变短)而增大的规律与波动理论严格一致，验证了模型的物理正确性。本研究可为声波勘探、医学超声、工程波动模拟等领域，提供一种高效高精度的数值工具。

Abstract: This paper proposes a high-accuracy numerical method for the three-dimensional wave equation based on polynomial particular solutions, and investigates the numerical simulation of the three-dimensional wave equation under different propagation velocities. The method employs polynomial particular solutions of various orders that satisfy the wave equation as basis functions to numerically simulate the wave equation. Numerical examples with different propagation velocities are selected for solution and comparison with the traditional finite difference method (FDM). The results show that this algorithm has higher accuracy and easier implementation than the traditional FDM. The rule that the calculation errors of both algorithms increase with the decrease of wave velocity (corresponding to the shortening of physical wavelength) is strictly consistent with the wave theory, which verifies the physical correctness of the model. This work provides an efficient and high-precision numerical tool for applications in seismic exploration, medical ultrasound, and general engineering wave propagation problems.