交错网格任意阶导数有限差分格式及差分系数推导

交错网格有限差分算法以其高效、精确、实用等优点在地震波数值模拟中得到广泛应用。目前交错网格有限差分的精度已达到时间4阶、空间2N阶；然而在求空间三次导数时，差分格式实际上并未达到所谓的2N阶精度，而是采用了低阶的差分格式及差分系数,这样有利于提高大尺度空间正演时的计算效率；但从计算精度的角度考虑，有必要推导出准确的满足2N阶精度的交错网格有限差分格式及差分系数，以得到更高精度的正演结果。笔者利用Taylor公式展开首次推导出了可导函数任意次导数的任意偶数阶精度的差分近似式及相应的差分系数，从而完善了常规高精度交错网格有限差分算法。采用新推导的交错网格有限差分格式得到的正演波形与解析解进行了对比，证明了新推导的差分格式的正确性，并与常规差分格式的正演波形进行了比较，结果显示,新推导出的交错网格有限差分格式模拟结果稳定性好，精度更高。

Staggered Grid Finite Difference Scheme and Coefficients Deduction of Any Number of Derivatives

Staggered grid finite difference algorithm is effective, accurate and practical, so it has wide application prospect and practical significance.So far, the common high-order scheme of staggered grid finite difference algorithm is 4-order temporal and 2N-order spatial accuracy.However, the FD scheme isn’t 2N-order accuracy actually when computing second or third spatial derivative.The authors deduce the FD scheme with 2N-order accuracy and corresponding coefficients of any number of derivatives of functions which have any number of derivatives forthefirsttime.So we can consummate conventional high-order staggered grid finite difference algorithm.We make a simulation of seismic response with conventional FD scheme and the new FD scheme respectively，and compared with analytic solution respectively. As a result, the new FD scheme is more stabilized and more accurate.