基于时空域交错网格有限差分法的应力速度声波方程数值模拟

A time-space domain dispersion-relationship-based staggered-grid finite-difference scheme for modeling the stress-velocity acoustic wave equation

目前应力速度声波方程数值模拟普遍采用时间二阶和空间2M阶交错网格差分法,相应的差分系数仅利用空间域频散关系和泰勒展开求解。但波动方程数值求解在时间和空间域同时进行,仅利用空间域频散关系计算差分系数,易产生数值频散,因而影响数值模拟精度。针对该问题,从差分离散波动方程和平面波理论出发,推导出了时间二阶、空间2M阶交错网格差分法的时空域频散关系,并进一步导出了基于时空域频散关系和泰勒展开的差分系数算法,该算法求解的差分系数随地震波的传播速度自适应变化。数值频散分析结果表明,新的差分系数算法能够有效减小数值频散进而提高模拟精度;稳定性分析结果表明,新的差分系数算法能够有效增强交错网格有限差分法的稳定性,使得该方法能采用更大的时间步长从而提高计算效率。层状介质模型和塔里木盆地典型复杂构造模型数值模拟实例进一步验证了基于新差分系数算法的交错网格有限差分法在提高模拟精度和计算效率方面的优越性。

Numerical simulations of the stress-velocity acoustic wave generally adopt a staggered grid difference with 2nd-order in time and 2M-order in space.The corresponding difference coefficients are calculated only based on the spatial domain dispersion relationship and the Taylor expansion.When the numerical solution of the wave equation is obtained in the time and space domains simultaneously,only the dispersion relationship in the space domain is used to calculate the difference coefficients.This easily causes numerical dispersion and affects the accuracy of the numerical simulation.To overcome this issue,we used the discrete difference wave equation and plane wave theory to derive the time-space domain dispersion relationship for calculating the staggered grid difference scheme of the second order in time and 2Mth order in space,and then developed an algorithm for calculating the difference coefficient based on the time-space domain dispersion relationship and the Taylor expansion.The so-obtained difference coefficient varies adaptively with the propagation speed of the seismic wave.Numerical dispersion analysis showed that the proposed algorithm can effectively reduce the numerical dispersion,thereby improving the simulation accuracy.Stability analysis showed that the proposed algorithm can effectively enhance the stability of the staggered grid difference scheme,making it can adopt a large time step to improve the calculation efficiency.Numerical simulation examples of a layered medium model and a complex structure model of the Tarim Basin further verified the superiority of the staggered grid finite difference scheme using the proposed difference coefficient algorithm in terms of simulation accuracy and calculation efficiency.