二维地震波场的组合型紧致有限差分数值模拟

地震波场数值模拟在地球物理勘探和地震学中具有重要的支撑作用.本文将组合型紧致差分格式用于声波和弹性波方程的数值模拟中.根据泰勒级数展开和声波方程，建立了位移场时间四阶离散格式，并将组合型紧致差分格式用于位移场空间导数的求取，然后对该差分格式进行了精度分析、误差分析、频散分析和稳定性分析.理论研究结果表明：①该差分格式为时间四阶、空间六阶精度，与常规七点六阶中心差分和五点六阶紧致差分相比，具有更小的截断误差和更高的模拟精度；②每个波长仅需要5.6个采样点，且满足稳定性条件的库郎数为0.792，可以使用粗网格和较大时间步长进行计算.所以该方法具有占用内存少、计算效率高和低数值频散等优势.最后，本文进行了二维各向同性完全弹性介质的声波和弹性波方程的数值模拟，实验结果表明本文提出的方法具有更高的计算精度，能够大幅度的节约计算量和内存需求，对于三维大尺度模型问题具有更好的适应性.

Numerical simulation of 2D seismic wave-field used combined compact difference scheme

The numerical simulation of seismic wave field carries great significance in geophysical exploration and seismology. In this paper, the combined compact difference scheme is used in the numerical simulation of acoustic equation and elastic wave equation. According to the Taylor series expansion and acoustic wave equation, the fourth-order discrete scheme of the acoustic equation is established, and the combined compact difference scheme is employed to calculate the spatial derivative of the displacement field, and then the accuracy, error, dispersion and stability of the difference scheme are analyzed. The results show that ① the difference scheme has a fourth-order temporal and sixth-order spatial accuracy, with smaller truncation error and higher simulation accuracy than the conventional 7-point 6-order center difference and 5-point 6-order compact difference. ② Each wavelength requires only 5.6 sampling points, and the Courant number meets the stability condition of being 0.792, so the coarse grid and the larger time step can be applied in calculation, with such merits as less memory, higher computational efficiency and lower numerical dispersion. Finally, the numerical simulation of the acoustic and elastic wave equations of the 2D isotropic and completely elastic media is carried out. The results show that the method proposed in this paper has higher accuracy in calculation, markedly reduces the volume of calculation and requirement of memory, and has better adaptability to large-scale 3D models.