【作者】 陈山；

【导师】 杨顶辉；

【作者基本信息】 清华大学 ， 数学， 2010， 博士

【摘要】 地震波传播正演模拟一直是地球物理与石油勘探领域的一个研究热点。本文首先提出了求解波动方程的一种新的数值方法——Runge-Kutta(龙格-库塔)方法。该方法的基本思想是:将二阶波动方程改写为一阶偏微分方程组,在空间方向上使用网格点上的位移、粒子速度及它们的梯度值的组合来近似逼近空间高阶偏导数,使一阶偏微分方程组转化为一个半离散的常微分方程组,然后对这个半离散的常微分方程组采用三阶或四阶Runge-Kutta方法进行时间推进计算,从而获得了求解波动方程的Runge-Kutta方法。本文还针对Runge-Kutta方法存在的不足进行改进,提出了一种加权Runge-Kutta方法,并将这两种方法统称为“Runge-Kutta型方法”。本文从理论分析和数值计算两个方面对Runge-Kutta型方法进行了系统的研究,主要包括以下研究内容:分别给出了求解波动方程的一维、二维和三维Runge-Kutta方法的具体实现步骤,并使用Fourier分析方法推导了一维、二维和三维Runge-Kutta方法的稳定性条件;对一维、二维和三维Runge-Kutta方法分别进行了误差和频散分析,并与经典的Lax-Wendroff修正方法和交错网格方法进行了数值精度和频散误差的比较;对不同介质中二维和三维的声波和弹性波的传播进行了数值模拟,并给出了模拟结果;给出了加权Runge-Kutta方法的实现步骤以及取不同加权参数时其稳定性条件;并用加权Runge-Kutta方法进行了不同介质中地震波传播的数值模拟,给出了模拟结果,同时分析了计算效率和存储效率。理论分析和数值实验结果表明,Runge-Kutta方法具有稳定性好,数值精度高,频散误差小的优点,并且在粗网格条件下能有效压制数值频散;加权Runge-Kutta方法可以进一步节省存储空间,更加有效地压制数值频散,从而可以通过使用更大的空间和时间步长以获得更快的计算速度和更小的存储量需求。因此,Runge-Kutta方法及其加权方法将在地球物理与石油勘探领域具有巨大的应用潜力。更多还原

【Abstract】 The forward modeling research of seismic wave propagation is one of the hotspot in the fields of Geophysics and petroleum exploration. In this thesis, we present a new numerical method, which are called Runge-Kutta method, to solve wave equations. We transform the second-order wave equations into a system of first-order partial differential equations and use not only the displacement and the particle-velocity of the grids, but also their gradients to approximate the high-order spatial derivatives, resulting that we get semi-discrete ordinary differential equations (ODEs). We then apply the Runge-Kutta method of order three or four to solve the semi-discrete ODEs and obtain our Runge-Kutta method for solving wave equations. In order to improve the Runge-Kutta method, we propose a weighted Runge-Kutta method. And we identically call the two methods“Runge-Kutta type method”.In this thesis, we systemically investigate the Runge-Kutta type method including theoretical analyses and numerical computations. The main research contents are listed as follows. Firstly, the construction of Runge-Kutta methods to solve 1D, 2D, and 3D wave equations are respectively described, and the stability conditions of the 1D, 2D, and 3D Runge-Kutta methods are derived by Fourier analysis method. Secondly, the numerical error and dispersion of the 1D, 2D, and 3D Runge-Kutta methods are respectively analyzed, and the numerical accuracy and dispersion error of the Runge-Kutta method are also compared with those of the classic Lax-Wendroff correction method and the staggered-grid method. Thirdly, 2D and 3D numerical simulation results of acoustic and elastic wave propagation in different media are presented. Fourthly, the implementation steps and stability conditions of the weighted Runge-Kutta method with different weights are given. Finally, numerical simulation results of the seismic wave propagation are shown by the weighted Runge-Kutta method, meanwhile, the computation and storage efficiencies are investigated. The results of the theoretical analyses and numerical computations indicate that the Runge-Kutta method has good stability, high numerical accuracy, and small dispersion error, and can effectively suppress the numerical dispersion on coarse grids. And the weighted Runge-Kutta method can further save the storage and suppress the numerical dispersion, so the coarser spatial and temporal grids can be used to acquire faster computational speed and smaller storage. Therefore, the Runge-Kutta method and its weighted method have great potentiality of applying in Geophysics and petroleum exploration.更多还原