Research progress on the numerical simulation of seismic wave motion based on spectral element method
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摘要:
基于谱元法的地震波动数值模拟已被广泛用于地震震源破裂、大规模地震波传播、区域复杂场地及工程结构（群）地震反应、地震层析成像等重要问题的研究及应用当中，是目前地震工程学、地震学和地球物理学等地震科技领域共同关注的前沿热点技术。早期发展的切比雪夫谱元法（CSEM）和勒让德谱元法（LSEM）更接近谱方法的域分解思路，其形式相对复杂且计算效率较低。目前广泛使用的是一种形式简洁的LSEM，其实施步骤和主要公式已经与有限元法完全一致，仅通过内置的高斯-洛巴托-勒让德（GLL）高精度数值积分保留着与谱方法之间的联系。谱元法的巨大成功不仅源于算法本身的高精度、规整性和灵活性，更是得益于以SPECFEM2D/3D，SPECFEM_GLOBE，SPEED等为代表的开源软件集成了复杂模拟所需的各项关键技术，包括三维复杂介质建模、震源模型数值实现、平面地震波输入、大规模并行计算、全球地震波模拟、伴随方法以及多尺度或不连续方法等。本文全面介绍了CSEM、LSEM、间断伽辽金谱元法（DG-SEM或DGM）、三角形单元谱元法、谱元法精度和稳定性方面的研究或应用进展，并详细阐述了谱元法在我国的发展历程以及我国学者关于谱元法研究与工程应用的学术贡献。谱元法可归属于有限元法范畴，其高阶单元具有优良的精度和稳定性，并能从理论上严格地导出集中质量矩阵。在地震波动领域各种形式的有限差分法和有限元法中，地震学和地球物理学的速度结构反演或震源参数反演对地震波到时、波形等细节比较敏感，通常需采用谱元法或高阶交错网格有限差分法等高精度方法。而地震工程领域主要关注不同工程结构、非线性岩土介质或者流体-固体多场耦合等情形下的力和变形，此时具有丰富单元库的有限元法常常更为有效。最后，考虑到二阶及以上谱单元的性能显著优于一阶有限单元，进一步研究不同地震工程问题的谱元解法具有重要意义，而且随着震源-路径-场地-结构（群）的地震灾害全过程模拟的日益发展，谱元法这种具有灵活单元阶次变化、宽频带模拟精度和高效并行能力的特殊高阶有限元法将会受到越来越多关注。
Abstract:The spectral element method （SEM）-based numerical simulation of seismic wave motion has been widely applied in the study of earthquake source rupture process, large-scale seismic wave propagation, seismic response of regional complex sites without/with engineering structures （systems）, seismic tomography, and so forth. This technique is currently a frontier hotspot of common concern in the fields of earthquake science and technology including earthquake engineering, seismology, geophysics, etc. Spectral element method, which is sometimes also termed as spectral finite element method （SPECFEM）, spectral element method, or hp-type element method, is a combination of spectral method and finite element method （FEM）. Hence, it shares the advantages of both the two methods, i.e., the high accuracy and fast convergence of spectral method, and the regularity and flexibility of FEM.
In early times, the Chebyshev spectral element method （CSEM） and Legendre spectral element method （LSEM） originated from the domain decomposition of spectral methods, and therefore they inherit the complicated formulations of the latter, in which each of the interpolation basis functions is a linear combination of Chebyshev or Legendre orthogonal polynomials. Consequently, both the methods are as accurate as the spectral methods, but their applications are severely limited by the cumbersome and inefficient multi-layer nested computational structure that is resulted from those basis functions. Nowadays, the most frequently-used SEM is a concise form of LSEM developed by Komatitsch et al. In this method, the early-used complicated basis functions are simplified to the Lagrange shape functions that are commonly adopted in FEM, and those orthogonal polynomial-based analytical Gauss-Lobatto-Legendre （GLL） quadrature formulae are replaced by a simple numerical list of the GLL point coordinates and integration weights. Specifically, the non-equally distributed GLL points serve as the element nodes and the GLL high-precision numerical integration formula is applied to calculate the element mass, stiffness matrices, etc. This configuration makes the LSEM enjoy the same solution procedure and computational formulations as that of FEM, but avoid the significant defects of the classical high-order finite element method, including the intrinsic numerical error of the high-order polynomial interpolation based on equally-spaced grid and the lower computational efficiency due to the high-order consistent mass matrix. In a word, this LSEM has actually become a high-performance lumped-mass high-order finite element method. In addition to the above methods, the family of non-conforming spectral element methods has also been broadly studied and successfully applied in many problems, making themselves an important branch of the SEM. By introducing the so-called Lagrange multiplier or interior penalty term as a glue to effectively connect spectral elements with quite different sizes, orders, shapes and so on, the non-conforming SEMs are more flexible and highly efficient in dealing with multi-scale or discontinuous problems, which appear frequently in large-scale or complicated seismic wave simulations.
The great success of SEM is not only due to the high accuracy, regularity and flexibility of the algorithm itself, but also attributed to those well-designed open-source SEM programs represented by SPECFEM2D/3D, SPECFEM_GLOBE, SPEED, etc. These programs have incorporated a variety of key technologies that are required in complex simulations, such as three-dimensional complex models, different seismic source models or plane wave input method, large-scale parallel computing, global simulation, adjoint method, multi-scale or discontinuous modeling and so on. In the field of earthquake engineering, the SEM has been applied to perform physics-based deterministic numerical simulation of strong ground motion and to realize the “end-to-end” seismic response analysis that is from the source rupture to engineering structures or even engineering systems. The massive simulation data is a good supplement to the insufficient strong ground motion records, and the modeling of seismic wave propagation in actual geolocial structures can compensate for the weak physical background of traditional ground motion prediction equations （GMPEs） or stochastic methods. These simulations, which have reached a certain level of reliability, bring new vitality to earthquake engineering research and application. In the fields of seismology or geophysics, the highly-efficient forward simulation of SEM has been combined with the adjoint method, enabling a simultaneous modeling of the seismic wave fields generated from a number of observation stations, thus the number of forward simulations required for an inversion process can be reduced to an acceptable level. In this way, the advanced full wave inversion （FWI） or seismic tomography technique has been practically used to investigate seismic source mechanisms and to reveal regional or even global velocity structures. Finally, the development of SEM in China is elaborated. The SEM was introduced into China around the year of 2000, and the related studies mainly focused on the basic performance of the method and some preliminary applications until early 2010s. In the past decade, the Chinese researchers have been conducting more and more innovative work on the SEM algorithms and various engineering applications, and some of the work has reached the research forefront of the world.
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引言
1935年Richter(1935，1958)在公布美国加州第一份地震目录时引入了地方性震级标度ML，1959年李善邦(1981)将Richter的原始震级公式引入我国，其表达式为
(1) 式中：Aμ为地动位移，单位为μm，Aμ=(AN+AE)/2，AN和AE分别为N-S分向和E-W分向S波(或Lg波)位移的最大振幅；R(Δ)为量规函数; Δ为震中距．李善邦(1981)结合我国当时常用的62型短周期地震仪器和基式中长周期地震仪器的特性，得到了R1(Δ)和R2(Δ)两个量规函数，其中R1(Δ)适用于短周期地震仪，R2(Δ)适用于中长周期地震仪(刘瑞丰等，2015)．根据《地震及前兆数字观测技术规范》中地震观测的要求，目前我国地震台网在计算地方性震级时使用的是R1(Δ)(中国地震局，2001)．R1(Δ)和R2(Δ)都是根据华北地区地震波的衰减特性得到的，与全国其它地区有一定差异．量规函数描述了地震波随震中距衰减的特性，与地壳构造紧密相关(陈培善，秦嘉政，1983；薛志照，1992)，因此，对区域性差异较大的地区，采用同一个量规函数显然是不合理的(严尊国等，1992，1995)．如果量规函数不准确，可能使各区测定的相同震级值的地震并不等同(陈培善，秦嘉政，1983)，故有必要建立我国分区地方性震级量规函数．
新中国成立后，我国先后建设了昆明、成都、兰州、南京、佘山、拉萨、广州和北京等一些国家地震台站；1966年以后，我国又陆续建设了北京、上海、沈阳、兰州、昆明和成都等6大区域地震台网以及西昌、呼和浩特、临汾、郑州、邯郸、太原、嘉祥、南京、汕头、大同、天津、唐山、银川和乌鲁木齐等小区域地震台网；到1978年，国家地震台站增至86个，全国区域地震台站增至435个；至2007年底，国家地震台网和31个省级地震台网正式运行的台站已达1007个．几十年来，国家地震台网和省级地震台网已经积累了大量的地震观测资料，为建立我国分区地方性震级量规函数的获取创造了有利的条件．
1. 资料收集
对于本项研究，地震观测数据的收集、整理是个较棘手的问题．2002年以前，我国省级地震台网一般使用模拟地震观测数据编辑地震观测报告，这些地震观测数据均保存在各省地震局．各省地震局数据管理的单位不同，数据格式也不同，特别是一些省地震局1990年以前的数据都是纸介质的地震观测报告，因此，我们组织全国31个省、自治区和直辖市地震局地震监测中心的业务人员利用一年多时间对1973年以来不同数据格式、不同存储介质的历史震相数据，按统一的数据格式进行录入和整理，共收集整理了全国31个省级地震台网1973—2002年1308个台站的震相数据，ML0.1以上的有效地震共计10万5282次，地震资料37万5744组．全国地震观测数据整合统计信息列于表 1，震中与台站分布如图 1所示．
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图 1 1973年1月—2002年12月全国ML≥0.1地震与台站分布图Figure 1. Distribution of ML≥0.1 earthquakes(dots)and seismic stations (triangles)in China from January 1973 to December 2002表 1 1973—2002年全国各省地震观测数据整合统计表Table 1. The statistics of the earthquake observation data in all the provinces of China from 1973 to 2002台网名称 地震次数 台站个数 资料组数 观测时段 安徽 3208 51 8268 1976—2002 福建 7534 55 47851 1989—2002 甘肃 4650 63 13537 1990—2000 广东 3592 50 3944 1990—2000 广西 3043 27 7492 1980—2002 海南 452 17 2256 2000—2002 北京、天津、河北 4614 163 8872 1989—2002 河南 2542 31 6406 1981—2002 黑龙江 1071 36 2531 1973—2002 湖北 2065 53 10508 1980—2002 湖南 350 18 1132 1987—2002 吉林 252 15 2561 1990—2001 江苏、上海 3831 33 13224 1982—2002 江西 400 7 748 1991—2002 辽宁 2037 50 11952 1980—2002 内蒙古东部 2280 23 10243 1990—2002 内蒙古中西部 2697 26 12289 1990—2002 宁夏 2858 17 11785 1990—2002 青海 4691 12 16385 1990—2002 山东 3936 52 24690 1975—2002 山西 7884 63 21402 1991—2000 陕西 1244 21 22202 1990—2002 重庆、四川 15300 92 37462 1990—2002 西藏 2363 26 2738 2001—2002 新疆 7238 54 33926 1990—2000 云南、贵州 14900 235 40684 1990—1999 浙江 250 18 656 1988—2002 注：2000年以前海南的地震观测数据属广东台网. 2. 计算方法
震级残差统计分析方法是陈培善和秦嘉政(1983)根据Christoskov等(1978)提出的方法加以改进得到的，首先用单台地震的震级值减去该地震的平均值得到单台地震的震级偏差，然后再绘出震级偏差随震中距的变化曲线，进而得出新的量规函数．
设Ne为地震次数，Ns为记录地震的台站个数，根据式(1)，求得第i次地震第j个台站的单台震级MLij(陈培善，秦嘉政，1983；陈运泰，刘瑞丰，2004)为
(2) 对第i次地震，其对所有台站的震级平均值MLij为
(3) 将式(2)减去式(3)，得到单台震级的偏差值ΔMLij为
(4) 根据单台震级的偏差值可得到震级偏差ΔMLij随震中距Δ的变化曲线，如果量规函数正确，则震级偏差随震中距的变化曲线就应该在0值附近摆动．曲线偏差值的负值为量规函数的校正值，将校正值加到量规函数R1(Δ)上，即可得到新的量规函数(陈培善，秦嘉政，1983；陈继锋等，2013)．
按照上述方法，依次计算每个省的震级偏差，绘制各省的震级偏差随震中距的变化曲线和量规函数曲线；在此基础上，将相邻省的震级偏差随震中距的变化曲线以及量规函数曲线进行对比，并依据我国地质构造情况和地方性震级的精度允许范围，将区别不大的省份合并为一个区域．鉴于内蒙古自治区占地面积较大，将其分为东部和中西部地区，然而经对比发现，这两个地区的差别并不大，可以合并．这样，最终将全国31个省份划分为5个大区，分别为东北与华北(黑龙江、吉林、辽宁、内蒙古、北京、天津、河北、山西、山东、河南、宁夏和陕西)、华南(福建、广东、广西、海南、江苏、上海、浙江、江西、湖南、湖北和安徽)、西南(云南、四川、重庆和贵州)、青藏(青海、西藏和甘肃)和新疆．从实用性和方便性出发，5大分区所对应的新的地方性震级量规函数依次表示为R11(Δ)，R12(Δ)，R13(Δ)，R14(Δ)和R15(Δ)．其中，东北与华北地区共497个台站，3万1415次地震，13万5033组震级资料;华南地区共329个台站，2万4725次地震，9万6079组震级资料；西南地区共327个台站，3万200次地震，7万8146组震级资料；青藏地区共101个台站，1万1704次地震，3万2660组震级资料；新疆地区共54个台站，7238次地震，3万3926组震级资料．
3. 计算结果及分析
选取各分区地震资料比较集中的范围进行计算，将得到的单台震级的偏差值进行曲线拟合，并按10 km的震中距间距进行平滑，震级偏差随震中距的变化曲线如图 2所示．可以看出：对于东北与华北、青藏地区，当震中距处于0—100 km时，偏差小于0，处于300—550 km时，偏差大于0(图 2a，d)；对于华南地区，当震中距处于450—550 km时，偏差大于0，曲线波动较大，其它则在0值附近上下波动(图 2b)；对于西南地区，当震中距处于0—50 km时，偏差小于0，处于200—300 km时，偏差大于0(图 2c)；对于新疆地区，当震中距处于0—230 km时，偏差小于0，处于300—450 km时，偏差大于0(图 2e)．
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图 2 单台震级偏差ΔMLij随震中距Δ的变化图(a)东北与华北地区;(b)华南地区;(c)西南地区;(d)青藏地区;(e)新疆地区 蓝色圆点为单台震级偏差值，红线为震级偏差平滑曲线，绿线为零线Figure 2. Variation of magnitude deviation ΔMLij of single station with epicentral distance Δ(a)Northeast-North China;(b)South China;(c)Southwest China;(d)Qinghai-Xizang region;(e)Xinjiang region The blue dots represent the local magnitude deviation of single station，the red curve represents smooth curve of local magnitude deviation，and the green curve represents zero curve图 3给出了各分区新量规函数与量规函数R1(Δ)的对比图，新的量规函数与震级偏差随震中距的变化曲线是相对应的．可以看出: 对于东北与华北、青藏地区，当震中距Δ处于0—100 km时，R11(Δ)>R1(Δ)，R14(Δ)>R1(Δ)，处于300—550 km时，R11(Δ)＜R1(Δ)，R14(Δ)＜R1(Δ)(图 3a，d)；对于华南地区，R1(Δ)与R12(Δ)两条曲线重合度最高，当Δ处于450—550 km时，R12(Δ)＜R1(Δ)(图 3b)；对于西南地区，当Δ处于0—50 km时，R13(Δ)>R1(Δ)，处于200—300 km时，R13(Δ)＜R1(Δ)(图 3c)；对于新疆地区，当Δ处于0—230 km时，R15(Δ)>R1(Δ)，处于300—450 km时，R15(Δ)＜R1(Δ)(图 3e)．
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图 3 5大分区的新量规函数与量规函数R1(Δ)对比图(a)东北与华北地区;(b)华南地区;(c)西南地区;(d)青藏地区;(e)新疆地区Figure 3. Comparison of new calibration functions with R1(Δ)(a)Northeast-North China;(b)South China;(c)Southwest China;(d)Qinghai-Xizang region;(e)Xinjiang region经计算所得的新量规函数列于表 2．在计算过程中，每个分区地震资料集中的震中距范围不一致，东北与华北、华南和青藏等3个地区集中在0—600 km，西南地区集中在0—300 km，新疆地区集中在0—550 km．在这些范围之外，因地震观测数据较少，计算得出的结果不可靠；但考虑到新量规函数的实用性以及地方性震级的定义，根据量规函数R1(Δ)和观测资料拟合曲线，将新量规函数的震中距范围扩展至0—1000 km．
表 2 新量规函数表Table 2. New calibration functionsΔ/km R11 R12 R13 R14 R15 0—5 1.9 1.8 2.0 2.0 2.0 10 2.0 1.9 2.0 2.1 2.1 15 2.2 2.1 2.1 2.2 2.2 20 2.3 2.2 2.2 2.3 2.3 25 2.5 2.4 2.4 2.5 2.5 30 2.7 2.6 2.6 2.6 2.6 35 2.9 2.8 2.7 2.8 2.8 40 2.9 2.9 2.8 2.9 2.8 45 3.0 3.0 2.9 3.0 2.9 50 3.1 3.1 3.0 3.1 3.0 55 3.2 3.2 3.1 3.2 3.1 60 3.3 3.3 3.2 3.2 3.2 70 3.3 3.3 3.2 3.2 3.2 75 3.4 3.4 3.3 3.3 3.3 85 3.3 3.3 3.3 3.4 3.3 90 3.4 3.4 3.4 3.5 3.4 100 3.4 3.4 3.4 3.5 3.4 110 3.5 3.5 3.5 3.6 3.6 120 3.5 3.5 3.5 3.6 3.6 130 3.6 3.6 3.6 3.7 3.6 140 3.6 3.6 3.6 3.7 3.6 150 3.7 3.7 3.7 3.8 3.7 160 3.7 3.7 3.7 3.7 3.7 170 3.8 3.8 3.8 3.8 3.8 180 3.8 3.7 3.8 3.8 3.8 190 3.9 3.8 3.9 3.9 3.9 200 3.9 3.9 3.9 3.9 3.9 210 3.9 4.0 3.9 4.0 3.9 220 3.9 4.0 3.9 4.0 4.0 230 4.0 4.1 4.0 4.1 4.0 240 4.1 4.1 4.0 4.1 4.0 250 4.1 4.2 4.0 4.1 4.1 260 4.1 4.2 4.1 4.1 4.1 270 4.2 4.2 4.2 4.2 4.2 280 4.2 4.3 4.1 4.1 4.1 290 4.3 4.4 4.2 4.2 4.2 300 4.2 4.4 4.3 4.2 4.3 310 4.3 4.5 4.4 4.3 4.4 320 4.3 4.4 4.4 4.3 4.4 330 4.4 4.5 4.5 4.4 4.4 340 4.4 4.5 4.5 4.4 4.4 350 4.4 4.5 4.5 4.5 4.5 360 4.5 4.6 4.5 4.5 4.5 370 4.5 4.6 4.5 4.4 4.5 380 4.5 4.6 4.6 4.5 4.5 390 4.5 4.6 4.6 4.5 4.5 400 4.6 4.7 4.7 4.5 4.6 420 4.6 4.7 4.7 4.6 4.7 430 4.6 4.7 4.8 4.7 4.7 440 4.6 4.7 4.8 4.75 4.8 450 4.6 4.7 4.8 4.75 4.8 460 4.6 4.7 4.8 4.75 4.8 470 4.7 4.7 4.8 4.8 4.8 500 4.8 4.7 4.8 4.8 4.8 510 4.8 4.8 4.9 4.9 4.9 530 4.8 4.8 4.9 4.9 4.9 540 4.8 4.8 4.9 4.9 4.9 550 4.8 4.8 4.9 4.9 4.9 560 4.9 4.9 4.9 4.9 4.9 570 4.8 4.9 4.9 4.9 4.9 580 4.9 4.9 4.9 4.9 4.9 600 4.9 4.9 4.9 4.9 4.9 610 5.0 5.0 5.0 5.0 5.0 620 5.0 5.0 5.0 5.0 5.0 650 5.1 5.1 5.1 5.1 5.1 700 5.2 5.2 5.2 5.2 5.2 750 5.2 5.2 5.2 5.2 5.2 800 5.2 5.2 5.2 5.2 5.2 850 5.2 5.2 5.2 5.2 5.2 900 5.3 5.3 5.3 5.3 5.3 1000 5.3 5.3 5.3 5.3 5.3 4. 量规函数的检验
为了检验新的地方性震级量规函数的精度和可用性，采用原量规函数R1(Δ)和新的分区量规函数R11(Δ)，R12(Δ)，R13(Δ)，R14(Δ)和R15(Δ)分别对1973—2002年ML≥0.1地震的地方性震级进行重新测定，并对单台震级偏差和标准误差进行对比．
4.1 单台震级偏差统计对比
利用5大分区的量规函数与量规函数R1(Δ)计算所得单台震级偏差的统计对比如图 4所示．可以看出：东北与华北、华南、青藏和新疆等4个地区使用新量规函数计算所得的单台震级偏差明显较R1(Δ)计算所得的单台震级偏差更加集中，且向ΔMLij=0靠拢；而西南地区的对比结果不明显．对西南地区进行数据统计，用R1(Δ)计算所得的单台震级偏差绝对值小于0.2的占65.3%，处于0.3—0.5的占28.6%，处于0.6—0.8的占5.2%，大于0.9的占0.8%；用R13(Δ)计算所得的单台震级偏差绝对值小于0.2的占66.2%，处于0.3—0.5的占28.4%，处于0.6—0.8的占4.8%，大于0.9的占0.5%，因此使用R13(Δ)比R1(Δ)计算所得的单台震级偏差更加集中．
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图 4 5大分区的新量规函数与量规函数R1(Δ)计算所得的单台震级偏差统计对比图(a)东北与华北地区;(b)华南地区;(c)西南地区;(d)青藏地区;(e)新疆地区Figure 4. The statistical comparisons of local magnitude deviation of single station by new calibration functions with those by R1(Δ)(a)Northeast-North China;(b)South China;(c)Southwest China;(d)Qinghai-Xizang region;(e)Xinjiang region总的看来，使用新量规函数R11(Δ)，R12(Δ)，R13(Δ)，R14(Δ)和R15(Δ)较量规函数R1(Δ)得到的单台震级偏差值更加集中，这说明使用新量规函数会减小单台震级偏差．
4.2 标准误差对比
标准误差是评判地方性震级优劣程度的一个依据(陈培善，秦嘉政，1983)．对第i次地震，其对所有台站震级的标准误差为
(5) 各区域所有地震对应的标准误差为
(6) 首先，用R1(Δ)计算本文所收集的所有地震的单台震级值，据式(5)和(6)得出各省的标准误差，进而得出各分区的标准误差S1；然后，用表 2中的新量规函数计算本文所收集的所有地震的单台震级值，进而得出各分区的标准误差S2，结果列于表 3．可以看出，两个标准误差的值为0.22—0.29，且S2＜S1，表明新量规函数计算得到的震级优于R1(Δ)计算得到的震级，故使用新的量规函数能够提高地方性震级的测定精度．
表 3 两种量规函数计算所得的震级标准误差Table 3. The magnitude standard deviations calculated by two kinds of calibration functions区域 S1 S2 东北与华北地区 0.25 0.24 华南地区 0.24 0.22 西南地区 0.26 0.25 青藏地区 0.26 0.25 新疆地区 0.29 0.28 5. 讨论与结论
本文选用1973—2002年全国31个省级地震台网的地震观测资料，根据震级残差分析统计方法，依次绘出各省的震级偏差随震中距的变化曲线和量规函数曲线；经相邻省之间进行相互对比，将全国分为东北与华北、华南、西南、青藏和新疆等5大分区，对应的量规函数依次记为R11(Δ)，R12(Δ)，R13(Δ)，R14(Δ)和R15(Δ)．从这5个新量规函数与量规函数R1(Δ)的对比可以看出，新旧量规函数的变化在0.1—0.2之间．
由于省级地震台网的观测资料主要集中在震中距Δ≤600 km的范围内，因此震中距Δ≤600 km的新量规函数较旧量规函数有一定变化，而600 km＜Δ≤1000 km的量规函数并没有变化．通过实际测定地方性震级表明，使用新量规函数较R1(Δ)计算所得的单台震级偏差更加集中，并向ΔMLij=0靠拢；而且使用新量规函数较R1(Δ)计算所得的地方性震级的标准误差有一定减小，这说明使用新的量规函数能够提高地方性震级的测定精度．
31个省级地震台网中心为本研究整理了1973年以来的地震观测资料，作者在此向所有参与资料整理的科技人员表示衷心的感谢．
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图 1 标量波情形下1—10阶谱元法的网格频散曲线（引自De Basabe，Sen，2007）
Figure 1. Grid dispersion curves of 1st- to 10th-order spectral elements in acoustic case （from De Basabe，Sen，2007）
1 高斯-洛巴托-勒让德高精度数值积分的节点和权系数
1 The nodes and weights of Gauss-Lobatto-Legendre high-precision numerical integration
GLL节点数 积分节点 积分权系数 2 ±1 1 3 0 1.333 333 333 333 333 3 ±1 0.333 333 333 333 333 3 4 ±0.447 213 595 499 957 9 0.833 333 333 333 333 4 ±1 0.166 666 666 666 666 7 5 0 0.711 111 111 111 111 1 ±0.654 653 670 707 977 2 0.544 444 444 444 444 5 ±1 0.100 000 000 000 000 0 6 ±0.285 231 516 480 645 1 0.554 858 377 035 486 2 ±0.765 055 323 929 464 7 0.378 474 956 297 847 0 ±1 0.066 666 666 666 666 7 7 0 0.487 619 047 619 047 6 ±0.468 848 793 470 714 2 0.431 745 381 209 862 7 ±0.830 223 896 278 567 0 0.276 826 047 361 565 9 ±1 0.047 619 047 619 047 6 8 ±0.209 299 217 902 478 9 0.412 458 794 658 703 8 ±0.591 700 181 433 142 3 0.341 122 692 483 504 4 ±0.871 740 148 509 606 6 0.210 704 227 143 506 1 ±1 0.035 714 285 714 285 7 9 0 0.371 519 274 376 417 2 ±0.363 117 463 826 178 2 0.346 428 510 973 046 3 ±0.677 186 279 510 737 7 0.274 538 712 500 161 7 ±0.899 757 995 411 460 2 0.165 495 361 560 805 5 ±1 0.027 777 777 777 777 8 10 ±0.165 278 957 666 387 0 0.327 539 761 183 897 6 ±0.477 924 949 810 444 5 0.292 042 683 679 683 8 ±0.738 773 865 105 505 0 0.224 889 342 063 126 4 ±0.919 533 908 166 458 9 0.133 305 990 851 070 1 ±1 0.022 222 222 222 222 2 注：积分节点和权系数的数目与GLL节点数相对应，数值相同、正负号不同的积分节点对应相同的积分权系数。 
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表 1 经典二阶中心差分格式用于谱元法时的稳定条件（De Basabe，Sen，2010）
Table 1 Stability criteria for classical second-order central-difference scheme applied in spectral element method （De Basabe，Sen，2010）
谱单元
阶次标量波情形
（SH波动）弹性波情形
（P-SV波动）谱单元
阶次标量波情形
（SH波动）弹性波情形
（P-SV波动）qEmax qdmax qEmax qdmax qEmax qdmax qEmax qdmax 1 0.709 0.709 0.816 0.816 5 0.071 4 0.608 0.082 3 0.700 2 0.288 0.577 0.333 0.666 6 0.051 6 0.608 0.059 5 0.700 3 0.164 0.593 0.189 0.684 7 0.039 0 0.608 0.044 9 0.700 4 0.104 0.604 0.120 0.697 8 0.030 4 0.607 0.035 0 0.699
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