Indirect methods of collocation: Trefftz‐Herrera collocation

A nonstandard collocation method (TH‐collocation) is presented, where collocation is used to construct specialized weighting functions instead of the solution itself, as it is usual, so that in this sense it is an indirect method. TH‐collocation is shown to be as accurate as standard collocation, but computationally far more efficient. The present article is the first of a series devoted to explore thoroughly collocation methods. The following classification of collocation methods is introduced: direct‐nonoverlapping; indirect‐nonoverlapping; direct‐overlapping; and indirect‐overlapping. Most of the effort reported in the literature has gone to direct‐nonoverlapping methods. The procedure presented in this article falls into the indirect‐nonoverlapping category and it is based on Trefftz‐Herrera formulation. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 709–738, 1999     

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

This article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines

Summary.   The collocation tension spline is considered as a numerical solution of a singularly perturbed two-point boundary value problem: . The collocation points are chosen as a generalization of the classical Gaussian points. Unlike the traditional approach, we employ the B-spline representation in the analysis. This leads to global quadratic convergence of the method for small perturbation parameters, and, for large values, the order of convergence is four. Received October 4, 1996 / Revised version received September 23, 1999 / Published online October 16, 2000     

Rational Chebyshev collocation method for solving higher‐order linear ordinary differential equations

A collocation method to find an approximate solution of higher‐order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor‐Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1130–1142, 2011     

Nonconforming spline collocation methods in irregular domains

This article studies a class of nonconforming spline collocation methods for solving elliptic PDEs in an irregular region with either triangular or quadrilateral partition. In the methods, classical Gaussian points are used as matching points and the special quadrature points in a triangle or quadrilateral element are used as collocation points. The solution and its normal derivative are imposed to be continuous at the marching points. The authors present theoretically the existence and uniqueness of the numerical solution as well as the optimal error estimate in H¹‐norm for a spline collocation method with rectangular elements. Numerical results confirm the theoretical analysis and illustrate the high‐order accuracy and some superconvergence features of methods. Finally the authors apply the methods for solving two physical problems in compressible flow and linear elasticity, respectively. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

High order compact solution of the one‐space‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving one‐space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivative of linear hyperbolic equation and collocation method for the time component. The main property of this method additional to its high‐order accuracy due to the fourth order discretization of spatial derivative, is its unconditionally stability. In this technique the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method produce a very efficient method for solving the one‐space‐dimensional linear hyperbolic equation. We compare the numerical results of this paper with numerical results of (Mohanty, 3 .© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Implicit Extension of Taylor Series Method with Numerical Derivatives

The Taylor series method is one of the earliest analytic‐numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of these algorithms is based on the approximate calculation of higher order derivatives using well‐known technique for the partial differential equations. The implicit extension based on a collocation term added to the explicit truncated Taylor series. This idea is different from the general collocation method construction, which led to the implicit R‐K algorithms [1].     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

High order implicit collocation method for the solution of two‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving the two dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component. The resulted method is unconditionally stable and solves the two‐dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A Cartesian‐grid collocation method based on radial‐basis‐function networks for solving PDEs in irregular domains

This paper reports a new Cartesian‐grid collocation method based on radial‐basis‐function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second‐ and fourth‐order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Finite‐difference schemes for transport‐dominated equations using special collocation nodes

We introduce finite‐difference schemes based on a special upwind‐type collocation grid, in order to obtain approximations of the solution of linear transport‐dominated advection‐diffusion problems. The method is well suited when the diffusion parameter is very small compared to the discretization parameter. A theory is developed and many numerical experiments are shown. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

In this article, we propose two meshless collocation approaches for solving time dependent partial differential algebraic equations (PDAEs) in terms of the multiquadric quasi‐interpolation schemes. In presenting the process of the solution, the error is estimated. Furthermore, the comparisons on condition numbers of the collocation matrices using different methods and the sensitivity of the shape parameter c are given. With the use of the appropriate collocation points, the method for PDAEs with index‐2 is improved. The results show that the methods have some advantages over some known methods, such as the smaller condition numbers or more accurate solutions for PDAEs which has an modal index‐2 or an impulse solution with index‐2. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 95–119, 2014     

Space‐time spectral collocation method for the one‐dimensional sine‐Gordon equation

In this article, we introduce a new space‐time spectral collocation method for solving the one‐dimensional sine‐Gordon equation. We apply a spectral collocation method for discretizing spatial derivatives, and then use the spectral collocation method for the time integration of the resulting nonlinear second‐order system of ordinary differential equations (ODE). Our formulation has high‐order accurate in both space and time. Optimal a priori error bounds are derived in the L²‐norm for the semidiscrete formulation. Numerical experiments show that our formulation have exponential rates of convergence in both space and time. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 670–690, 2015     

Quartic B‐spline collocation algorithms for numerical solution of the RLW equation

The collocation method based on quartic B‐spline interpolation is studied for numerical solution of the regularized long wave (RLW) equation. The time‐split RLW equation is also solved with the quartic B‐spline collocation method. Numerical accuracy is tested by obtaining the single solitary wave solution. Then, interaction, undulation and evolution of solitary waves are studied. Solutions are compared with available results. Conservation quantities are computed for all test experiments. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations

In this paper, we consider a piecewise linear collocation method for the solution of a pseudo‐differential equation of order r=0, ?1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three‐point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low‐order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low‐order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]³) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N^?(2?r)/2). Note that, in contrast to well‐known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderón–Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd.     

A collocation approach for solving linear complex differential equations in rectangular domains

In this paper, a collocation method is presented to find the approximate solution of high‐order linear complex differential equations in rectangular domain. By using collocation points defined in a rectangular domain and the Bessel polynomials, this method transforms the linear complex differential equations into a matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. The proposed method gives the analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and the comparisons are made with existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on a computer using a program written in MATLAB v7.6.0 (R2008a). Copyright © 2012 John Wiley & Sons, Ltd.     

A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon

A quadrature Galerkin scheme with the Bogner–Fox–Schmit element for a biharmonic problem on a rectangular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known that a product Gaussian quadrature with at least three‐points is required to guarantee optimal order convergence in Sobolev norms. In this article, optimal order error estimates are proved for a scheme based on the product two‐point Gaussian quadrature by establishing a relation with an underdetermined orthogonal spline collocation scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Study and analysis of a two‐dimensional nonconservative fractional order aerosol transport equation

The shifted Legendre collocation method is used to solve the two‐dimensional fractional order aerosol equation with initial and boundary conditions. The solution profile of the equation is presented graphically for different cases. The important feature of the article is graphical exhibitions of the effect of the size of the aerosol particles and also the temporal derivative on the solution profile. The salient feature of the article is the demonstration of lower variation of mass concentration with the change in time level in fractional order systems than that in integer order system. A drive has been taken towards the tabular and pictorial presentations of a comparison of the numerical solution of our proposed method with an analytical solution of an existing problem through error analysis which conforms super‐linearly convergence rate of the proposed method to validate its efficiency and effectiveness.     

Legendre spectral collocation method for volterra-hammerstein integral equation of the second kind

This paper is concerned with obtaining the approximate solution for VolterraHammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function ω(x) = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L~2 norm and L~∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.