| 求解弹性波方程的辛RKN格式 |
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| 引用本文： | 刘少林,李小凡,汪文帅,朱童.求解弹性波方程的辛RKN格式[J].地球物理学报,2015,58(4):1355-1366. |
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| 作者姓名： | 刘少林 李小凡 汪文帅 朱童 |
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| 作者单位： | 1. 中国科学院地质与地球物理研究所, 中国科学院地球与行星物理重点实验室, 北京 100029;2. 宁夏大学数学与计算机学院, 银川 750021;3. 中国石油化工股份有限公司石油物探技术研究院, 南京 210014 |
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| 基金项目： | 国家自然科学基金(41174047,40874024和41204041)资助. |
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| 摘 要： | 将弹性波方程变换至Hamilton体系,构造适用于弹性波模拟的高效显式二阶辛Runge-Kutta-Nystrm(RKN)格式,运用根数理论得到此格式的阶条件方程组.通过给定系数的限定条件,得到方程的对称解.为了使时间离散误差达到极小,提出数值频率与真实频率比较,通过Taylor展开,得到关于辛系数的限定方程,求解方程组得到最小频散辛RKN格式.对比分析时间演进方程的稳定性,得到使库朗数达到极大值的限定方程,求解方程组得到最稳定辛RKN格式.发现此两种格式为同一格式.新得到的辛RKN格式不依赖于空间离散方法,为了对比的需要,选取有限差分法进行空间离散.在频散、稳定性分析中,与常见辛格式对比,从理论上分析了本文提出的格式在数值频散压制、稳定性提升等方面的优势,数值实验进一步证实了理论分析的正确性.
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| 关 键 词： | 弹性波方程 辛RKN格式 稳定性条件 频散关系 有限差分法 |
| 收稿时间： | 2014-07-15 |
A symplectic RKN scheme for solving elastic wave equations |
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| LIU Shao-Lin,LI Xiao-Fan,WANG Wen-Shuai,ZHU Tong.A symplectic RKN scheme for solving elastic wave equations[J].Chinese Journal of Geophysics,2015,58(4):1355-1366. |
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| Authors: | LIU Shao-Lin LI Xiao-Fan WANG Wen-Shuai ZHU Tong |
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| Institution: | 1. Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China;2. School of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China;3. SINOPEC Geophysical Research Institute, Nanjing 210014, China |
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| Abstract: | The construction of low dispersive and strong stable numerical schemes is critical for the investigation of wave propagation. Solving seismic wave equations in the time domain, the dispersion and stability are related to the temporal and spatial discretizations. Here we focus on the temporal discretization and develop a high-efficient time integration scheme for the elastic wave equation.Following the transformation of the elastic seismic wave equation into a Hamiltonian system, an explicit second-order symplectic Runge-Kutta-Nyström (RKN) scheme is proposed for elastic wave modeling. The order conditions are obtained by the rooted trees theory. To obtain symmetric solutions, we assign a constraint condition to the symplectic coefficients. In order to minimize the error in temporal discretization, we propose a method that compares numerical angular frequency with real frequency, and obtain a constraint equation based on Taylor expansion. A symplectic RKN scheme with minimized numerical dispersion is developed by solving the order conditions associating with constraint equations. Through analyzing the time advance equation, we construct another constraint equation to allow stability limit to attain its maximal value, and then another symplectic RKN scheme with the same symplectic coefficients as the first scheme is developed. The optimization process for determining the coefficients of the RKN scheme is independent of spatial discretizations. For the ease of comparison, we simply choose finite difference method to approximate spatial derivatives. Generally, the numerical dispersion amount is lower than the conventional second-order schemes, and is slightly larger than the high-order schemes. The maximal Courant number for the new scheme is the largest compared to those for the conventional schemes. The largest time increment for the new scheme is nearly two times as large as those for the conventional second-order schemes, and is about 1.5 times as large as those for the high-order ones. In a practical simulation, all these schemes discussed in the paper consume approximately the same computer memory, but the numerical accuracy and computational speed are quite different. The numerical accuracy and complexity of our scheme indicate that it may be a good candidate for the balance of numerical accuracy and complexity.Both theoretical analyses and numerical experiments show that our scheme is superior to conventional symplectic schemes in some aspects including suppressing dispersion and increasing stability range. The optimization procedure proposed in the paper may be used to define the coefficients in other symplectic or non-symplectic schemes. Though striking numerical results are obtained by our scheme with a finite difference operator, a low-dispersive spatial operator is still needed to be combined with our scheme to form a more powerful tool for wave simulations. |
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| Keywords: | Elastic wave equation RKN Dispersion relation Stability condition Finite difference method |
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