二维黏滞声波方程的优化组合型紧致有限差分数值模拟

Numerical simulation of 2D visco-acoustic wave equation with an optimized combined compact difference scheme

根据泰勒级数展开和黏滞声波方程,建立了位移场时间二阶离散格式,并将组合型紧致差分方法用于位移场空间导数的求取,然后对该差分格式进行了模拟精度、频散关系和稳定性分析,并基于频散关系保持的思想,探讨了组合型紧致差分格式的优化。理论研究结果表明:①三点六阶组合型紧致差分格式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低数值频散;②黏滞声波方程差分格式的频散关系和稳定性不仅与空间网格大小和时间步长有关,而且与介质品质因子和地震波主频有关;③优化后的差分格式比优化前数值波数更接近真波数,更有利于压制数值频散、提高计算效率。最后,利用PML边界条件,对均匀和Marmousi模型进行了黏滞声波方程的数值模拟和波场特征分析,验证了本文提出的方法能够适用于复杂介质的数值模拟,并具有较高的模拟精度和计算效率。

The numerical simulation of seismic wavefield has great significance in geophysical exploration and seismology.In this paper,a combined compact difference scheme is proposed for the numericalsimulation of visco-acoustic wave equation.First the second-order discrete scheme of the displacement field is established according to the Taylor series expansion and visco-acoustic wave equation, and a combined compact difference scheme is used to calculate the spatial derivative of the displacement field.Then the accuracy,dispersion and stability of the difference scheme are analyzed.Finally the optimization of the combination compact difference scheme is discussed based on the idea of dispersion-relation-preserving.The following understandings are obtained from'our theoretical research:A.3-point 6-order combined compact difference scheme has smaller truncation errors and lower simulation dispersion than the conventional 7-point 6-order center difference and 5-point 6-order compact difference;B.The dispersion relation and the stability of the visco-acoustic wave equation difference scheme are not only related to the space grid size and time step,but also to the media quality factor and the seismic dominant frequency; G.The numerical wavenumber of the optimized combination compact difference scheme is more close to the true wavenumber than that before optimization,which is more beneficial to suppress the numerical dispersion and improve the calculation efficiency.The numerical simulation and wavefield of visco-acoustic wave equation are analyzed on uniform and Marmousi models based on the perfectly matched layer (PML)boundary condition.Numerical simulation results show that the proposed scheme has higher accuracy and computational efficiency in complex media.