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.

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.

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.

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.

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.

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.

EXPONENTIAL FOURIER COLLOCATION METHODS FOR SOLVING FIRST-ORDER DIFFERENTIAL EQUATIONS

In this paper, a novel class of exponential Fourier collocation methods （EFCMs） is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation meth- ods. We discuss in detail the connections of EFCMs with trigonometric Fourier colloca- tion methods （TFCMs）, the well-known Hamiltonian Boundary Value Methods （HBVMs）, Gauss methods and Radau IIA methods. It turns out that the novel EFCMs are an es- sential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of first- order ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM（2,2） as an illustrative example. The numerical experiments using EFCM（2,2） are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM（2,2）.

Moving collocation methods for time fractional differential equations and simulation of blowup

A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.

Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed

For a simple model of a scalar wave equation with a random wave speed,Gottlieb and Xiu[Commun.Comput.Phys.,3(2008),pp.505-518]employed the generalized polynomial chaos(gPC)method and demonstrated that when uncertainty causes the change of characteristic directions,the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues.Consequently,a consistent method of imposing the boundary conditions is proposed and its convergence is established under the assumption that the expansion coefficients decay fast asymptotically.In this work,we investigate stochastic collocation methods for the same type of scalar wave equation with random wave speed.It will be demonstrated that the rate of convergence depends on the regularity of the solutions;and the regularity is determined by the random wave speed and the initial and boundary data.Numerical examples are presented to support the analysis and also to show the sharpness of the assumptions on the relationship between the random wave speed and the initial and boundary data.An accuracy enhancement technique is investigated following the multi-element collocation method proposed by Foo,Wan and Karniadakis[J.Comput.Phys.,227(2008),pp.9572-9595].

Convergence Analysis on Stochastic Collocation Methods for the Linear Schr¨odinger Equation with Random Inputs

In this paper,we analyse the stochastic collocation method for a linear Schr¨odinger equation with random inputs,where the randomness appears in the potential and initial data and is assumed to be dependent on a random variable.We focus on the convergence rate with respect to the number of collocation points.Based on the interpolation theories,the convergence rate depends on the regularity of the solution with respect to the random variable.Hence,we investigate the dependence of the stochastic regularity of the solution on that of the random potential and initial data.We provide sufficient conditions on the random potential and initial data to ensure the smoothness of the solution and the spectral convergence.Finally,numerical results are presented to support our analysis.

Analysis and Application of Stochastic Collocation Methods for Maxwell’s Equations with Random Inputs

In this paper we develop and analyze the stochastic collocation method for solving the time-dependent Maxwell’s equations with random coefficients and subject to random initial conditions.We provide a rigorous regularity analysis of the solution with respect to the random variables.To our best knowledge,this is the first theoretical results derived for the standard Maxwell’s equations with random inputs.The rate of convergence is proved depending on the regularity of the solution.Numerical results are presented to confirm the theoretical analysis.

Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel p（t, s） = （t - s）^-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233：938 950], the error analysis for this approach is carried out for 0 〈 μ 〈 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-ype but also establish the error estimates under a more general regularity assumption on the exact solution.

SUPERGEOMETRIC CONVERGENCE OF SPECTRAL COLLOCATION METHODS FOR WEAKLY SINGULAR VOLTERRA AND FREDHOLM INTEGRAL EQUATIONS WITH SMOOTH SOLUTIONS

A spectral collocation method is proposed to solve Volterra or Fredholm integral equations with weakly singular kernels and corresponding integro-differential equations by virtue of some identities. For a class of functions that satisfy certain regularity conditions on a bounded domain, we obtain geometric or supergeometric convergence rate for both types of equations. Numerical results confirm our theoretical analysis.

Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

In this paper,a Chebyshev-collocation spectral method is developed for Volterra integral equations(VIEs)of second kind with weakly singular kernel.We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity.The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points.The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials,approximation theory for orthogonal polynomials,and the operator theory.The spectral rate of convergence for the proposed method is established in the L^(∞)-norm and weighted L^(2)-norm.Numerical results are presented to demonstrate the effectiveness of the proposed method.

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.

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources.The singular source term is represented by theδ-function.For the approximation of theδ-function,the direct projection method is used that was proposed in[6].Theδ-function is constructed in a consistent way to the derivative operator.Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method.Theδ-function with the spectral method is highly oscillatory but yields good results with small number of collocation points.The results are compared with those computed by the second order finite difference method.In modeling general hyperbolic equations with a non-stationary singular source,however,the solution of the linear scalar wave equation with the nonstationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme.The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

Nonlinear Stochastic Galerkin and Collocation Methods:Application to a Ferromagnetic Cylinder Rotating at High Speed

The stochastic Galerkin and stochastic collocation method are two state-ofthe-art methods for solving partial differential equations(PDE)containing random coefficients.While the latter method,which is based on sampling,can straightforwardly be applied to nonlinear stochastic PDEs,this is nontrivial for the stochastic Galerkin method and approximations are required.In this paper,both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder.This model can be used for designing solid-rotor induction machines in various machining tools.A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties.Implementation issues of the stochastic Galerkin method are addressed and a numerical comparison of the computational cost and accuracy of both methods is performed.The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy,however at a higher computational cost.

A Comparative Study of Stochastic Collocation Methods for Flow in Spatially Correlated Random Fields

Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years.Similar to Monte Carlo methods,the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it.The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method,Smolyak method,Stroud 2 or 3 cubature method,and adaptive Stroud method.Another type of collocation method,the probabilistic collocation method(PCM),has also been proposed and applied to flow in porous media.In this paper,we discuss these methods in terms of their accuracy,efficiency,and applicable range for flow in spatially correlated random fields.These methods are compared in details under different conditions of spatial variability and correlation length.This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency.The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods.Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.

Collocation Methods for A Class of Volterra Integral Functional Equations with Multiple Proportional Delays

In this paper,we apply the collocation methods to a class of Volterra integral functional equations with multiple proportional delays(VIFEMPDs).We shall present the existence,uniqueness and regularity properties of analytic solutions for this type of equations,and then analyze the convergence orders of the collocation solutions and give corresponding error estimates.The numerical results verify our theoretical analysis.