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.

THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS

This paper applies the singular integral operators, singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels, which are established in [1], to give a unified framework for various collocation methods of numerical solutions of singular integral equations with Cauchy kernels. Under the framework, the coincidence of the direct quadrature method and the indirect quadrature method is very simple and obvious.

Error Control Strategies for Numerical Integrations in Fast Collocation Methods

We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates.

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.

The Adaptive Wavelet Collocation Method and Its Application in Front Simulation

The adaptive wavelet collocation method （AWCM） is a variable grid technology for solving partial differential equations （PDEs） with high singularities. Based on interpolating wavelets, the AWCM adapts the grid so that a higher resolution is automatically attributed to domain regions with high singularities. Accuracy problems with the AWCM have been reported in the literature, and in this paper problems of efficiency with the AWCM are discussed in detail through a simple one-dimensional （1D） nonlinear advection equation whose analytic solution is easily obtained. A simple and efficient implementation of the AWCM is investigated. Through studying the maximum errors at the moment of frontogenesis of the 1D nonlinear advection equation with different initial values and a comparison with the finite difference method （FDM） on a uniform grid, the AWCM shows good potential for modeling the front efficiently. The AWCM is also applied to a two-dimensional （2D） unbalanced frontogenesis model in its first attempt at numerical simulation of a meteorological front. Some important characteristics about the model are revealed by the new scheme.

Wavelet Collocation Method for Solving Elliptic Singularly Perturbed Problem

Wavelet collocation method is used to solve an elliptic singularly perturbed problem with two parameters. The B-spline function is used as a single mother wavelet, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problem and the result shows the reliability and efficiency of the method.

Fast Multiscale Collocation Methods for a Class of Singular Integral Equations

This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.

Error Estimation and Assessment of an Approximation in a Wavelet Collocation Method

This article describes how to assess an approximation in a wavelet collocation method which minimizes the sum of squares of residuals. In a research project several different types of differential equations were approximated with this method. A lot of parameters must be adjusted in the discussed method here. For example one parameter is the number of collocation points. In this article we show how we can detect whether this parameter is too small and how we can assess the error sum of squares of an approximation. In an example we see a correlation between the error sum of squares and a criterion to assess the approximation.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.

Integrated interpretation of dual frequency induced polarization measurement based on wavelet analysis and metal factor methods

In mineral exploration, the apparent resistivity and apparent frequency （or apparent polarizability） parameters of induced polarization method are commonly utilized to describe the induced polarization anomaly. When the target geology structure is significantly complicated, these parameters would fail to reflect the nature of the anomaly source, and wrong conclusions may be obtained. A wavelet approach and a metal factor method were used to comprehensively interpret the induced polarization anomaly of complex geologic bodies in the Adi Bladia mine. Db5 wavelet basis was used to conduct two-scale decomposition and reconstruction, which effectively suppress the noise interference of greenschist facies regional metamorphism and magma intrusion, making energy concentrated and boundary problem unobservable. On the basis of that, the ore-induced anomaly was effectively extracted by the metal factor method.

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.

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

In this paper,a deep collocation method(DCM)for thin plate bending problems is proposed.This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning.Besides,the proposed DCM is based on a feedforward deep neural network(DNN)and differs from most previous applications of deep learning for mechanical problems.First,batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries.A loss function is built with the aim that the governing partial differential equations(PDEs)of Kirchhoff plate bending problems,and the boundary/initial conditions are minimised at those collocation points.A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters.In Kirchhoff plate bending problems,the C^1 continuity requirement poses significant difficulties in traditional mesh-based methods.This can be solved by the proposed DCM,which uses a deep neural network to approximate the continuous transversal deflection,and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

Probabilistic small signal stability analysis of power system with wind power and photovoltaic power based on probability collocation method

Recently, with increasing improvements in the penetration of wind power and photovoltaic power in the world, probabilistic small signal stability analysis(PSSSA) of a power system consisting of multiple types of renewable energy has become a key problem. To address this problem, this study proposes a probabilistic collocation method(PCM)-based PSSSA for a power system consisting of wind farms and photovoltaic farms. Compared with the conventional Monte Carlo method, the proposed method meets the accuracy and precision requirements and greatly reduces the computation; therefore, it is suitable for the PSSSA of this power system. Case studies are conducted based on a 4-machine 2-area and New England systems, respectively. The simulation results show that, by reducing synchronous generator output to improve the penetration of renewable energy, the probabilistic small signal stability(PSSS) of the system is enhanced. Conversely, by removing part of the synchronous generators to improve the penetration of renewable energy, the PSSS of the system may be either enhanced or deteriorated.

ELASTO-PLASTICITY ANALYSIS BASED ON COLLOCATION WITH THE MOVING LEAST SQUARE METHOD

A meshless approach based on the moving least square method is developed for elasto-plasticity analysis,in which the incremental formulation is used.In this approach,the dis- placement shape functions are constructed by using the moving least square approximation,and the discrete governing equations for elasto-plastic material are constructed with the direct collo- cation method.The boundary conditions are also imposed by collocation.The method established is a truly meshless one,as it does not need any mesh,either for the purpose of interpolation of the solution variables,or for the purpose of construction of the discrete equations.It is simply formu- lated and very efficient,and no post-processing procedure is required to compute the derivatives of the unknown variables,since the solution from this method based on the moving least square approximation is already smooth enough.Numerical examples are given to verify the accuracy of the meshless method proposed for elasto-plasticity analysis.

FRACTURE CALCULATION OF BENDING PLATES BY BOUNDARY COLLOCATION METHOD

Fracture of Kirchhoff plates is analyzed by the theory of complex variables and boundary collocation method. The deflections, moments and shearing forces of the plates are assumed to be the functions of complex variables. The functions can satisfy a series of basic equations and governing conditions, such as the equilibrium equations in the domain, the boundary conditions on the crack surfaces and stress singularity at the crack tips. Thus, it is only necessary to consider the boundary conditions on the external boundaries of the plate, which can be approximately satisfied by the collocation method and least square technique. Different boundary conditions and loading cases of the cracked plates are analyzed and calculated. Compared to other methods, the numerical examples show that the present method has many advantages such as good accuracy and less computer time. This is an effective semi_analytical and semi_numerical method.

Application of Boundary Collocation Method to Two-Dimensional Wave Propagation

Boundary Collocation Method （BCM） based on Eigenfunction Expansion Method （EEM）, a new numerical method for solving two-dimensional wave problems, is developed. To verify the method, wave problems on a series of beaches with different geometries are solved, and the errors of the method are analyzed. The calculation firmly confirms that the results will be more precise if we choose more rational points on the beach. The application of BCM, available for the problems with irregular domains and arbitrary boundary conditions, can effectively avoid complex calculation and programming. It can be widely used in ocean engineering.

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.

Simulation of electrically driven jet using Chebyshev collocation method

The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver 'ddaskr' is used to solve the ODEs and post-stabilization is executed at the end of each step.Results show the distributions of radius,linear charge density,stretching ratio and also the horizontal velocity at a time point.Meanwhile,the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.

MHD Falkner-Skan flow of Maxwell fluid by rational Chebyshev collocation method

The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.

Laguerre-Gauss collocation method for initial value problems of second order ODEs

This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. Numerical results demonstrate its high efficiency.