Review of Collocation Methods and Applications in Solving Science and Engineering Problems

The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations.This paper provides a comprehensive review of collocation methods and their applications,focused on elasticity,heat conduction,electromagnetic field analysis,and fluid dynamics.The merits of the collocation method can be attributed to the need for element mesh,simple implementation,high computational efficiency,and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method.Beginning with the fundamental principles of the collocation method,the discretization process in the continuous domain is elucidated,and how the collocation method approximation solutions for solving differential equations are explained.Delving into the historical development of the collocation methods,their earliest applications and key milestones are traced,thereby demonstrating their evolution within the realm of numerical computation.The mathematical foundations of collocation methods,encompassing the selection of interpolation functions,definition of weighting functions,and derivation of integration rules,are examined in detail,emphasizing their significance in comprehending the method’s effectiveness and stability.At last,the practical application of the collocation methods in engineering contexts is emphasized,including heat conduction simulations,electromagnetic coupled field analysis,and fluid dynamics simulations.These specific case studies can underscore collocation method’s broad applicability and effectiveness in addressing complex engineering challenges.In conclusion,this paper puts forward the future development trend of the collocation method through rigorous analysis and discussion,thereby facilitating further advancements in research and practical applications within these fields.

A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.

Effects of Collocational Congruency,Frequency and L2 Proficiency on the Processing of English Adjective-Noun Collocations

The present study investigates the effects of congruency and frequency on adjective-noun collocational processing for Chinese learners of English at two proficiency levels based on the data obtained in an online acceptability judgment task.The subject pool of this research included 60 English majors studying at a university in China;30 were selected as a higher-proficiency group and 30 as a lower-proficiency group according to their Vocabulary Levels Test(Schmitt et al.,2001)scores and their self-reported proficiency in English.The experimental materials were programmed to E-prime 2.0 and included six types of collocations:(1)15 high-frequency congruent collocations,(2)15 low-frequency congruent collocations,(3)15 high-frequency incongruent collocations,(4)15 low-frequency incongruent collocations,(5)15 Chinese-only items,and(6)75 unrelated items for baseline data.The collected response times(RTs)and accuracy rates data were statistically analyzed by the use of an ANOVA test and pairwise comparisons through SPSS 16.0 software.The results revealed that:(1)the adjective-noun collocational processing of Chinese English learners is influenced by collocational frequency,congruency and L2 proficiency;(2)the processing time is affected by the interaction of congruency and frequency;and(3)the interactive effect of L2 proficiency in conjunction with congruency and frequency also influences the processing quality.

Isogeometric Least-Squares Collocation Method with Consistency and Convergence Analysis

This paper presents the isogeometric least-squares collocatio method,which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense.The number of collocation points employed in IGA-L can be larger than that of the unknowns.Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods.First,a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution.Second,IGA-L method is more flexible and more stable,because the number of collocation points in IGA-L is variable.Third,IGA-L is convergent in some cases of singular parameterization.Moreover,the consistency and convergence analysis are also developed in this paper.

A Numerical Investigation Based on Exponential Collocation Method for Nonlinear SITR Model of COVID-19

In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.

基于Triple-Collocation多源地表温度数据融合方法研究

为充分利用多源地表温度数据,提高区域尺度下地表温度精度,改善地表温度遥感产品的适用性。以淮河流域为研究区,Triple-Collocation(TC)方法为基础,采用最小二乘法,由中分辨率光谱成像仪(MODIS)地表温度与风云三号可见光红外扫描辐射计(FY-3C/VIRR)地表温度遥感产品,结合地面观测站实测数据,研究多源数据融合方法。将淮河流域39个气象站点分为融合站点与验证站点,由融合站点采用反距离插值(IDW)算法得到区域连续地表温度数据,与FY-3C/VIRR地表温度数据、MODIS地表温度数据,融合得到淮河流域地表温度。与地面验证站点对比分析表明,融合后的地表温度,其R^(2)、RMSE和MAE分别为0.98、1.10 K、0.88 K,优于MODIS地表温度的0.92、2.59 K、2.18 K和FY-3C地表温度产品的0.84、6.68 K、5.35 K,精度得到了显著的提高。淮河流域不同月份的温度变化季节性规律明显,且变化主要出现在中部及东部地区,最高温度变化值都达到了20 K以上。

BITS: an efficient transport solver based on a collocation method with B-spline basis

A B-spline Interpolation Transport Solver(BITS) based on a collocation method is developed. It solves transport equations as a generalized interpolation problem, taking the first-order accuracy in time and the second-order accuracy in space along with a predictor–corrector or under-relaxation iteration method. Numerical tests show that BITS can solve one-dimensional transport equations for tokamak plasma more accurately without additional computation cost, compared to the finite difference method transport solver which is widely used in existing tokamak transport codes.

Finite Element Orthogonal Collocation Approach for Time Fractional Telegraph Equation with Mamadu-Njoseh Polynomials

Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.

Least-squares reverse time migration in visco-acoustic media based on symplectic stereo-modeling method

The authors proposed a symplectic stereo-modeling method(SSM)in the Birkhoffian dynam-ics and apply it to the visco-acoustic least-squares reverse time migration(LSRTM).The SSM adopts ste-reo-modeling operator in space and symplectic Runge-Kutta scheme in time,resulting in great ability in suppressing numerical dispersion and long-time computing.These advantages are further confirmed by numerical dispersion analysis,long-time computation test and computational efficiency comparison.After these theoretical analyses and experiments,acoustic and visco-acoustic LSRTM are tested and compared between SSM method and the conventional symplectic method(CSM)using the fault and marmousi models.Meanwhile,dynamic source encoding and exponential decay moving average gradients method are adopted to reduce the computation cost and improve the convergence rate.The imaging results show that LSRTM based on visco-acoustic wave equations effectively takes into account the influence of viscosity can therefore compensate for the amplitude attenuation.Besides,SSM method not only has high numerical accuracy and computational efficiency,but also performs effectively in LSRTM.

A Comparative Study on Verb-Noun Collocation Errors of Chinese EFL Learners with Different Language Proficiency

This study presents a comparative analysis of verb-noun collocation errors made by Chinese EFL learners with different language proficiency.Linguistic data from two sub-corpora of CLEC(Chinese Learner English Corpus,Gui&Yang,1999)consist of the compositions written freely by Chinese senior high school students(ST2)and the 3rd-and 4th-year English majors of Chinese universities(ST6).The study results show that four types of verb-noun collocation errors are frequently made due to incorrect L1 translation,misuse of relative synonyms,misuse of delexical verbs,and misuse of part of speech;the number of errors decreases with the development of language proficiency,especially errors by incorrect L1 translation and relative synonyms.And for both ST2 and ST6 EFL learners,errors resulted from incorrect L1 translation account for nearly a half with the same high-frequency words.It is also found that ST6 learners’competence in collocation and synonym identification doesn’t grow with the increase of their vocabulary,as is shown in the case of overusing delexical verbs in collocation,the same error made by ST2 learners,who have a significantly weaker command of delexical verbs.Implications of the findings are also discussed with the purpose of raising the awareness of collocation pedagogy and enhancing EFL learners’collocation competence.

基于Triple Collocation方法的土壤湿度误差分析

土壤湿度作为水循环中重要的驱动参数之一,对天气变化有着显著影响。遥感技术的发展促使土壤湿度的大范围动态性观测变为可能,但对其误差的准确估计仍需进一步的研究。利用ASCAT散射计、AMSR-E辐射计反演得到的2种卫星遥感土壤湿度数据以及ERA-Interim土壤湿度再分析资料,通过三重组合(Triple Collocation,TC)方法得到了研究区域(15°N^55°N,73°E^135°E) 3种土壤湿度数据的误差方差和信噪比估计,并结合MODIS土地覆盖类型数据分析了3种土壤湿度数据的误差特征。结果表明:植被覆盖会影响遥感土壤湿度的TC误差方差估计;从TC误差方差估计值来看,ERA土壤湿度精度最高,AMSR-E精度次之,ASCAT精度最低;从信噪比来看,ASCAT土壤湿度信噪比最高,ERA的信噪比低于ASCAT高于AMSR-E,AMSR-E信噪比最低;通过研究区MODIS土地覆盖类型数据与TC结果的分析可知TC结果多分布在草原、农田和裸地,TC结果比较符合客观实际情况。

小步长低阶Collocation方法轨道积分性能研究

介绍了Collocation方法的原理及误差构成,以Collocation方法在逐历元滤波轨道解算中的应用为切入点对小步长低阶Collocation方法轨道积分的性能进行了研究。分别就二体问题及真实GPS星座卫星轨道为对象进行了一系列验算,得到了一些有益结论:就积分方法本身性能而言,Collocation积分方法较传统数值积分法精度要高且稳定性要好;在一定长度的积分步长范围内,将小步长低阶Collocation方法用于滤波轨道解算是可行且有效的。但在积分步长很小的情形下,从计算效率方面考虑,不推荐使用Collocation方法;利用小步长低阶Collocation方法取代大步长高阶方法能提高效率且可抑制蚀现象引起的积分结果发散。经试验表明:选用积分步长为900s的4阶方法是合适的。

Collocation-PECE算法在低轨卫星精密定轨中的应用

在精密轨道确定中,PECE算法已成为目前最通用的轨道积分算法。利用Collocation方法的外推公式和内插公式可构造一种新的PECE算法,新算法以两种方式进行二阶微分方程的求解:将二阶微分方程降为一阶微分方程然后积分以及直接积分二阶微分方程。分别采用这两种类型的算法求解二体问题和低轨卫星的受摄轨道,并将计算结果与Adams-PECE算法及一般的PECE算法的积分结果进行了比较分析。结果表明,这两种类型的Collo-cation-PECE算法在实际轨道积分中较其它几种算法更有效。

基于统计的常用词搭配(Collocation)的发现方法

常用词搭配 (collocation)是指一些常用的用来表达某些事情或事物的短语 ,一般是二元组或三元组等。常用词搭配的自动发现在自然语言处理中起着很重要的作用 ,它能够丰富词典的容量 ,提高系统的性能。本文提出 4种判断一个二元组是否是常用词搭配的方法 ,并对各种方法的结果进行了比较。然后在已知二元组的基础上 ,提出了统计和发现三元或多元组的方法。这种利用二元组来构造三元组的方法 ,比统计所有三元组的计算量大大减少。

Least-Squares及Galerkin谱元方法求解环形区域内的泊松方程

为研究基于Least-Squares变分及Galerkin变分两种形式的谱元方法的求解特性,推导了极坐标系中采用两种变分方法求解环形区域内Poisson方程时对应的弱解形式,采用Chebyshev多项式构造插值基函数进行空间离散,得到两种谱元方法对应的代数方程组,由此分析了系数矩阵结构的特点。数值计算结果显示:Least-Squares谱元方法为实现方程的降阶而引入新的求解变量,使得代数方程组形式更为复杂,但边界条件的处理比Galerkin谱元方法更为简单;两种谱元方法均能求解极坐标系中的Poisson方程且能获得高精度的数值解,二者绝对误差分布基本一致;固定单元内的插值阶数时,增加单元数可减小数值误差,且表现出代数精度的特点,误差降低速度较慢,而固定单元数时,在一定范围内数值误差随插值阶数的增加而减小的速度更快,表现出谱精度的特点;单元内插值阶数较高时,代数方程组系数矩阵的条件数急剧增多,方程组呈现病态,数值误差增大,这一特点限制了单元内插值阶数的取值。研究内容对深入了解两种谱元方法在极坐标系中求解Poisson方程时的特点、进一步采用相关分裂算法求解实际流动问题具有参考价值。

Idioms,Collocations,Formulaic Phrases and Second Language Acquisition

Idioms,collocations,and formulaic phrases are three factors playing the important role in the second language acquisition(SLA).This essay firstly deals with the definitions of these concepts,then it illustrates the importance of them to SLA on two perspectives—idioms,collocations,and formulaic phrases respectively and they regarded as a whole.Therefore,in this way,the essay may lead to the answer the question“How are idioms,collocations,and formulaic phrases important to SLA?”

一种改进的collocation方法及其在动态定价问题中的应用

涉及多个状态变量的动态定价问题往往难以求解。基于此,引入多维插值与张量积,提出一种改进的collocation方法用于求解多状态变量的动态优化问题,并从理论上分析了改进后的collocation方法的收敛性。首先,将改进的collocation方法应用于求解多个状态变量的动态经济问题;给出了改进后的collocation方法的实际运用,应用于解决有限记忆的动态定价问题;利用状态变量的切比雪夫格点与其估计值构造出时序的状态转移,并依据状态转移模拟最优价格路径。通过数值分析表明,改进后的collocation方法比线性二次逼近法更快的收敛于稳态。

Study on the Collocation between Small Garden Ornaments and Other Elements

Starting with the basic elements of small garden ornaments, this research discusses its collocation art with other landscape elements such as plants, water, stones and environment, in order to offer useful references for the garden planning in a city.

THE ACCURACY COMPARISON BETWEEN CHEBYSHEV-τ METHOD AND CHEBYSHEV COLLOCATION METHOD

This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in plane Poiseuille flow for both methods are presented and compared in detail. It is acknowledged that the Chebyshev collocation method has higher precision than the other one, especially for near netural situation.

基于collocation方法的Taylor-Culick模型流动稳定性的特征向量分析

对于固体火箭发动机中产生的流动不稳定现象,需要利用简化模型进行分析,目前最符合实际的简化模型为Taylor-Culick流模型。采用稳定性理论对轴对称的Taylor-Culick模型进行数值计算时,由于对称轴处径向坐标为零而带来奇性,以往的数值计算方法难以处理。采用一种配置微分矩阵的方式来改进算法,避免柱坐标所带有的奇性对数值计算的影响。对固体火箭发动机Taylor-Culick流动模型进行了局部稳定性计算,得到了一致的结果。分析了特征向量随计算参数的变化,发现频率、雷诺数与特征向量的形态变化有直接关系,且特征向量的变化代表着特定情况下流动区域振荡发生的范围与形式,这些流动不稳定的局部特征从细节上反映了整体流动对应于不同流动参数的状态。