基于最小范数优化交错网格有限差分系数的波动方程数值模拟

在采用有限差分法进行波动方程数值模拟时，其固有的数值频散现象影响计算结果的精度。已有常系数优化方法，大多是在给定误差阈值条件下通过求解满足最宽波数覆盖范围的差分系数压制数值频散，但这会导致较小波数区间的频散误差较大，造成波场传播过程中显著的误差积累效应。为此，提出了一种新的声波方程交错网格优化有限差分正演模拟方法。首先基于L₁范数在波数域建立空间一阶导数的目标函数，然后采用交替方向乘子法（ADMM）求解交错网格有限差分系数。数值频散曲线对比表明，在万分之一的误差容限条件下，ADMM算法在中低波数域对频散误差的控制效果更好。均匀介质模型和复杂模型的数值实验证明，基于不同范数的优化方法中，L₁范数对误差积累的控制效果更优。

Numerical simulation of wave equations based on minimum-norm optimization of staggered-grid finite-difference coefficients

When the finite difference method is used for numerical simulation of wave equations, the inhe-rent numerical dispersion affects the accuracy of calculation results. Most of the existing constant coefficient optimization approaches suppress the numerical dispersion by solving the difference coefficient that satisfies the broadest wavenumber cove-rage under a given error threshold. This, however, increases the dispersion error in a small wavenumber interval, resulting in a significant error accumulation effect during the wavefield propagation. In view of this, this paper proposes a new finite difference forward simulation method for staggered grid optimization of acoustic equations. First, the objective function of the spatial first derivative is established with L₁ norm in the wavenumber domain, and then the staggered-grid finite-difference coefficients are solved using the alternating direction multiplier method (ADMM). The comparison of the numerical dispersion curves shows that the ADMM has a better control effect on the dispersion error in the low and middle wavenumber domains at the error tolerance of one ten-thousandth. The numerical experiments on homogeneous and complex models show that the L₁ norm has better control over the error accumulation among diffe-rent norms for optimization.