GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of highorder accuracy. To attain high accuracy in time discretization, the fourth-order AdamsBashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Hermite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation （nonlinear convection- diffusion problem） and KdV equation（single solitary and 2-solitary wave problems）, where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.

Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers Equations

The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplifed Jacobi operational matrix, we produce the diferentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary diferential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.

GENERALIZED JACOBI SPECTRAL GALERKIN METHOD FOR FRACTIONAL-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin method.The basis functions for the provided method are selected generalized Jacobi functions(GJFs),which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately constructed.The developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-norm.Numerical results indicate the usefulness of the proposed method.

Optimal error estimates for Fourier spectral approximation of the generalized KdV equation

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.

Q-factor estimation in CMP gather and the continuous spectral ratio slope method

The attenuation factor or quality factor(Q-factor or Q) has been used to measure the energy attenuation of seismic waves propagating in underground media. Many methods are used to estimate the Q-factor. We propose a method to calculate the Q-factor based on the prestack Q-factor inversion and the generalized S-transform. The proposed method specifies a standard primary wavelet and calculates the cumulative Q-factors; then, it finds the interlaminar Q-factors using the relation between Q and offset(QVO) and the Dix formula. The proposed method is alternative to methods that calculate interlaminar Q-factors after horizon picking. Because the frequency spectrum of each horizon can be extracted continuously on a 2D time–frequency spectrum, the method is called the continuous spectral ratio slope(CSRS) method. Compared with the other Q-inversion methods, the method offers nearly effortless computations and stability, and has mathematical and physical significance. We use numerical modeling to verify the feasibility of the method and apply it to real data from an oilfield in Ahdeb, Iraq. The results suggest that the resolution and spatial stability of the Q-profile are optimal and contain abundant interlaminar information that is extremely helpful in making lithology and fluid predictions.

Jacobi-Sobolev Orthogonal Polynomialsand Spectral Methods for Elliptic Boundary Value Problems

Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.

Structural Reliability Assessment by a Modified Spectral Stochastic Meshless Local Petrov-Galerkin Method

This study presents a new tool for solving stochastic boundary-value problems. This tool is created by modify the previous spectral stochastic meshless local Petrov-Galerkin method using the MLPG5 scheme. This modified spectral stochastic meshless local Petrov-Galerkin method is selectively applied to predict the structural failure probability with the uncertainty in the spatial variability of mechanical properties. Except for the MLPG5 scheme, deriving the proposed spectral stochastic meshless local Petrov-Galerkin formulation adopts generalized polynomial chaos expansions of random mechanical properties. Predicting the structural failure probability is based on the first-order reliability method. Further comparing the spectral stochastic finite element-based and meshless local Petrov-Galerkin-based predicted structural failure probabilities indicates that the proposed spectral stochastic meshless local Petrov-Galerkin method predicts the more accurate structural failure probability than the spectral stochastic finite element method does. In addition, generating spectral stochastic meshless local Petrov-Galerkin results are considerably time-saving than generating Monte-Carlo simulation results does. In conclusion, the spectral stochastic meshless local Petrov-Galerkin method serves as a time-saving tool for solving stochastic boundary-value problems sufficiently accurately.

广义概率密度演化方程的Chebyshev拟谱法

概率密度演化方法(probability density evolution equation,PDEM)为非线性随机结构的动力响应分析提供了新的途径.通过PDEM获得结构响应概率密度函数(probability density function,PDF)的关键步骤是求解广义概率密度演化方程(generalized probability density evolution equation,GDEE).对于GDEE的求解通常采用有限差分法,然而,由于GDEE是初始条件间断的变系数一阶双曲偏微分方程,通过有限差分法求解GDEE可能会面临网格敏感性问题、数值色散和数值耗散现象.文章从全局逼近的角度出发,基于Chebyshev拟谱法为GDEE构造了全局插值格式,解决了数值色散、数值耗散以及网格敏感性问题.考虑GDEE的系数在每个时间步长均为常数,推导了GDEE在每一个时间步长内时域上的序列矩阵指数解.由于序列矩阵指数解形式上是解析的,从而很好地克服了数值稳定性问题.两个数值算例表明,通过Chebyshev拟谱法结合时域的序列矩阵指数解求解GDEE得到的结果与精确解以及Monte Carlo模拟的结果非常吻合,且数值耗散和数值色散现象几乎可以忽略.此外,拟谱法具有高效的收敛性且序列矩阵指数解不受CFL (Courant-Friedrichs-Lewy)条件的限制,因此该方法具有良好的数值稳定性和计算效率.

广义Rosenau-Kawahara方程的有效谱方法

针对广义Rosenau-Kawahara方程提出了Legendre dual-Petrov-Galerkin谱方法,并基于对角化技巧,构建了快速有效算法。在此基础上研究了单个孤立波的传播、守恒律及波的生成等物理现象。数值结果验证了所提算法的有效性。

四阶混合边值问题的广义Jacobi-Petrov-Galerkin谱方法

发展了矩形区域上的四阶混合边值问题的广义Jacobi-Petrov-Galerkin谱方法,利用广义Jacobi多项式对模型问题的精确解进行数值展开,设计了有效的数值算法.数值结果验证了该算法的有效性和高精度.

THE LARGE TIME CONVERGENCE OF SPECTRAL METHOD FOR GENERALIZED KURAMOTO-SIVASHINSKY EQUATIONS

In this paper we use the spectral method to analyse the generalized Kuramoto-Sivashinsky equations. We prove the existence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estimation between spectral approximate solution and exact solution on large time.

Periodic spectral characteristics of seismicity before strong earthquakes and their application

Periodic spectral characteristics of earthquake activity in the seismic strengthening areas of 24 ewthquakes withM≥ 6.0 are studied by the maximum entropy spectral method whose superiority is tested. Then the follow resultshave been obtained : ① The periodic spectra of seismic activity in seismic strengthening areas are different indifferent stage in earthquake-generating processes. Long periodic spectra and short ones coexist in normal stage,while only short ones (on average, 43% of long ones) exist and long ones disappear prior to ear’thquakes. ② Theappearing time of short period before earthquakes has some relations with magnitude. The result shows thatdecades or even one hundred years is the common value for a great earthquake of M=8.0, 30 years for one withmagnitude about 7 and 20-30 years for a strong quake of M=6.0. For the same magnitude earthquakes in differentregions the appearing time is also different. For example, it is longer in North China than that in the western pan ofChina. Then the characteristics are preliminarily explained applying the strong body earthquake-generating model.Applying the maximum entropy spectral method, the idea of tendency prediction for strong and great earthquakesis suggested and used into practice. for example. the tendency predictions of the Wuding earthquake with M=6.5and the Lijiang earthquake of M=7.0 in Yunnan Province got some positive effects. So a new method of tendencyprediction of M≥6.0 earthquakes is offered.

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

Analysis of Nonlinear Stochastic Systems with Jumps Generated by Erlang Flow of Events

In this paper we consider the stochastic systems with jumps (random impulses) generated by Erlang flow of events that lead to discontinuities in paths. These systems may be used in various applications such as a control of complex technical systems, financial mathematics, mathematical biology and medicine. We propose to use a spectral method formalism to the probabilistic analysis problem for the stochastic systems with jumps. This method allows to get a solution of the analysis problem in an explicit form.

基于不同海域海况特点的通用型FPSO疲劳损伤分析

由于西非海域、墨西哥湾海域和巴西海域的海况特点存在明显差异,通用型FPSO在三个海域海况下的疲劳损伤呈现出较大差别,对通用型FPSO进行基于不同海域海况特点的疲劳损伤分析,结果表明,对于通用型FPSO,垂向弯矩对于船体结构疲劳强度的影响要比转矩的影响大得多,设计时应充分考虑垂向弯矩的影响;对于主船体关键热点,各海域不同系泊方式下的热点损伤均具有明显差异,其中多点系泊西非海域的疲劳损伤较大,巴西海域次之,墨西哥湾海域最小,内转塔系泊对应的疲劳损伤大于外转塔系泊,设计时应根据通用型FPSO的实际运行海域与系泊方式重点考虑损伤较大的海域与系泊方式。

基于PIGD-IRK的大规模时滞电力系统特征值高效计算方法

电力系统控制回路中固有的时滞会影响控制器的性能,甚至危害系统的稳定运行。为此,提出了一种基于无穷小生成元隐式龙格-库塔部分离散化(PIGD-IRK)的特征值计算方法,用以分析时滞对大规模电力系统小干扰稳定性的影响。首先,分析了制约无穷小生成元隐式龙格-库塔离散化(IGD-IRK)方法计算效率的瓶颈;然后,应用部分谱离散化思想大幅度降低其构造的无穷小生成元离散化矩阵的维数;接着,利用离散化矩阵的结构特点将其从阶梯型分块矩阵相似变换为常数矩阵的逆和分块上三角矩阵的乘积,从而可在特征值计算过程中充分利用离散化矩阵和系统增广状态矩阵的稀疏特性。最后,通过四机两区域系统和2个实际电网的计算结果验证了所提方法的准确性和高效性。

基于LR-GST变换的稳定层间Q值提取方法

为了解决常规谱比法不稳定及等效Q值储层识别精度低的问题,在基于Lucy-Richardson算法的广义S变换(LR-GST)的基础上,改进了谱比法计算Q值的公式,同时构建反演目标函数将等效Q值转换为层间Q值。结果表明:Q值求取精度与噪声比率、求取方法等因素有关,含噪比率不同每种方法的准确性也差异较大。通过合成数据及实际应用表明,本文方法的分辨率及聚集性更高,方法的稳定性更好,得到的Q值在含油气区域异常明显。

基于谱分解的气藏识别技术与应用

地震波传播穿过含油气砂岩时,高频能量发生明显衰减,研究地震波的这种衰减特征可以为油气检测提供有效信息。本文利用三类含气砂岩模型,针对完全弹性介质和盖层为弹性介质、储层为衰减介质两种情形,利用基于波动理论的反射率法分别正演模拟其相应的叠前地震记录,并利用广义S变换对模拟记录进行分频处理,分析不同频率时AVO特征。研究表明,弹性介质条件下AVO响应曲线不受频率影响;盖层为弹性介质、储层为衰减介质时不同频率的AVO响应曲线不同,随频率增加地震波衰减增大。据此对S油田实际资料进行分析,结果表明含气层表现出异常高衰减特性,利用谱分解技术可以有效预测气层的分布。

基于改进的广义S变换求取地层品质因子Q值

地层的品质因子Q值对衡量地震波传播过程中的能量衰减以及地震资料的处理、解释有重要意义。在实际生产中求取品质因子最实用的方法是频谱比法,但传统的频谱比法面临时窗选取等问题。本文利用改进的广义S变换时频特性及与傅里叶谱相联系的特性,提取地层上、下界面对应的瞬时频谱,并通过拟合振幅比与频率的关系得到地层的品质因子Q值。数值模拟及实际资料处理均证明了该方法的有效性。