贴体坐标系二维声波方程SBP有限差分法的稳定性分析

有限差分方法(Finite Difference Method,FDM)是波动方程正演数值模拟领域应用最为广泛的方法之一,然而,当模拟区域不规则或者地表起伏不平时,规则网格有限差分法求解波动方程会产生阶梯状近似,影响模拟的精度。借助贴体网格技术,将不规则的物理区域转换为规则的计算域,给出了贴体坐标系下的二维声波方程及其二阶精度的分部求和(Summation by Parts,SBP)有限差分离散格式，采用Fourier谱分析方法分析了该离散格式的稳定性,得到了贴体网格二维声波方程SBP有限差分方法的稳定性条件。数值实验结果表明：①当时间采样间隔的选取满足稳定性条件时,贴体网格SBP有限差分的数值计算过程是稳定的；②与贴体网格中心差分方法相比，贴体网格SBP有限差分方法的稳定性更好。

Traditionally,finite difference method is widely used as a fast and accurate method for numerical simulation and migration of wave equation.However,finite difference method faces obstacles when there are surface topography and irregular interfaces.The staircase approximations caused by regular grids influence the accuracy of finite difference method.Boundary-conforming grids by elliptic method offer an optimal alternative for finite difference wavefield simulation in irregular regions.By such grids,the calculations of spatial derivatives are transformed by a chain rule into those in the regular computational space,where traditional finite difference schemes are still applicable.The two-dimensional acoustic wave equation is reformulated in boundary-conforming grids.The Summation-by-Parts (SBP) finite difference scheme with second order accuracy is used for discretization.The stability of the discretization formula is discussed by Fourier spectral method to obtain the stability condition of the SBP finite difference scheme for two-dimensional acoustic wave equation in boundary-conforming grids.A numerical test shows that the accuracy and effectiveness can be guaranteed by reasonably choosing the parameters in numerical simulation,such as the temporal and spatial intervals.Comparisons between the numerical simulations of SBP and central finite difference method in boundary-conforming grids are performed,which shows that SBP finite difference scheme is more stable.

国家重点基础研究发展计划(973计划)(2013CB228603)、国家自然科学基金(41174119)和中石油物探新方法新技术研究(2014A-3609)项目共同资助。