L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and $(2?\alpha )$ order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.     

hp-Legendre–Gauss collocation method for impulsive differential equations

The original Legendre–Gauss collocation method is derived for impulsive differential equations, and the convergence is analysed. Then a new hp-Legendre–Gauss collocation method is presented for impulsive differential equations, and the convergence for the hp-version method is also studied. The results obtained in this paper show that the convergence condition for the original Legendre–Gauss collocation method depends on the impulsive differential equation, and it cannot be improved, however, the convergence condition for the hp-Legendre–Gauss collocation method depends both on the impulsive differential equation and the meshsize, and we always can choose a sufficient small meshsize to satisfy it, which show that the hp-Legendre–Gauss collocation method is superior to the original version. Our theoretical results are confirmed in two test problems.     

Exponential Runge–Kutta methods for delay differential equations

This paper deals with convergence and stability of exponential Runge–Kutta methods of collocation type for delay differential equations. It is proved that these kinds of numerical methods converge at least with their stage order. Moreover, a sufficient condition of the numerical stability is provided. Finally, some numerical examples are presented to illustrate the main conclusions.     

A note on the satisfaction of the boundary conditions for Chebyshev collocation methods in rectangular domains

The way boundary conditions are imposed when applying Chebyshev collocation methods to Poisson and biharmonic-type problems in rectangular domains is investigated. It is shown that careful selection of the number of collocation points leads to a linear system ofn linearly independent equations inn unknowns.     

Fast methods for the iterative solution of linear elliptic and parabolic partial differential equations involving 2 space dimensions

Within the last decade, attention has been devoted to the introduction of several fast computational methods for solving the linear difference equations which are derived from the finite difference discretisation of many standard partial differential equations of Mathematical Physics.

In this paper, the authors develop and extend an exact factorisation technique previously applied to parabolic equations in one space dimension to the implicit difference equations which are derived from the application of alternating direction implicit methods when applied to elliptic and parabolic partial differential equations in 2 space dimensions under a variety of boundary conditions.     

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection–diffusion and pseudo-parabolic equations.     

Adaptive Low-Rank Approximation of Collocation Matrices

This article deals with the solution of integral equations using collocation methods with almost linear complexity. Methods such as fast multipole, panel clustering and ℋ-matrix methods gain their efficiency from approximating the kernel function. The proposed algorithm which uses the ℋ-matrix format, in contrast, is purely algebraic and relies on a small part of the collocation matrix for its blockwise approximation by low-rank matrices. Furthermore, a new algorithm for matrix partitioning that significantly reduces the number of blocks generated is presented. Received August 15, 2002; revised September 20, 2002 Published online: March 6, 2003     

A B-spline collocation method for solving the incompressible Navier-Stokes equations using an ad hoc method: the Boundary Residual method

Collocation methods using piece-wise polynomials, including B-splines, have been developed to find approximate solutions to both ordinary and partial differential equations. Such methods are elegant in their simplicity and efficient in their application. The spline collocation method is typically more efficient than traditional Galerkin finite element methods, which are used to solve the equations of fluid dynamics. The collocation method avoids integration. Exact formulae are available to find derivatives on spline curves and surfaces. The primary objective of the present work is to determine the requirements for the successful application of B-spline collocation to solve the coupled, steady, 2D, incompressible Navier-Stokes and continuity equations for laminar flow. The successful application of B-spline collocation included the development of ad hoc method dubbed the Boundary Residual method to deal with the presence of the pressure terms in the Navier-Stokes equations. Historically, other ad hoc methods have been developed to solve the incompressible Navier-Stokes equations, including the artificial compressibility, pressure correction and penalty methods. Convergence studies show that the ad hoc Boundary Residual method is convergent toward an exact (manufactured) solution for the 2D, steady, incompressible Navier-Stokes and continuity equations. C¹ cubic and quartic B-spline schemes employing orthogonal collocation and C² cubic and C³ quartic B-spline schemes with collocation at the Greville points are investigated. The C³ quartic Greville scheme is shown to be the most efficient scheme for a given accuracy, even though the C¹ quartic orthogonal scheme is the most accurate for a given partition. Two solution approaches are employed, including a globally-convergent zero-finding Newton's method using an LU decomposition direct solver and the variable-metric minimization method using BFGS update.     

A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations

A non-uniform Haar wavelet based collocation method has been developed in this paper for two-dimensional convection dominated equations and two-dimensional near singular elliptic partial differential equations, in which traditional Haar wavelet method produces oscillatory solutions or low accurate solutions. The main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations. This process is done by discretizing space variables with non-uniform Haar wavelets. To confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods, finite element method, finite difference method and polynomial collocation method. Numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes.     

Analysis of a Chebyshev-based backward differentiation formulae and relation with Runge–Kutta collocation methods

The construction of a class of backward differentiation formulae based on intra-step Chebyshev–Gauss–Lobatto nodal points, suitable for the approximate numerical integration of initial-value problems of first-order ordinary differential equations, is presented. Formulae of this new family are A ₀-stable and L(α)-stable for any orders, and, particularly, for orders 1 and 2 they are L-stable. Regions of absolute stability and stability measures make this class very promising. We prove that this family of methods may be considered as Runge–Kutta collocation methods where the abscissae are obtained from the Chebyshev–Gauss–Lobatto points.     

Quadratic spline quasi-interpolants and collocation methods

Univariate and multivariate quadratic spline quasi-interpolants provide interesting approximation formulas for derivatives of approximated functions that can be very accurate at some points thanks to the superconvergence properties of these operators. Moreover, they also give rise to good global approximations of derivatives on the whole domain of definition. From these results, some collocation methods are deduced for the solution of ordinary or partial differential equations with boundary conditions. Their convergence properties are illustrated and compared with finite difference methods on some numerical examples of elliptic boundary value problems.     

A Fast Spectral Subtractional Solver for Elliptic Equations

The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N ² log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.     

Iterated Fast Collocation Methods for Integral Equations of the Second Kind

In this paper a new iteration technique is proposed based on fast multiscale collocation methods of Chen et al. (SIAM J Numer Anal 40:344–375, 2002) for Fredholm integral equations of the second kind. It is shown that an additional order of convergence is obtained for each iteration even if the exact solution of the integral equation is non-smooth, the kernel of the integral operator is weakly singular and the matrix compression is implemented. When the solution is smooth, this leads to superconvergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the method.     

Comparison of collocation methods for the solution of second order non-linear boundary value problems

This article concerns the application of cubic spline collocation tau-method for solving non-linear second order ordinary differential equations. Three collocation methods [Taiwo, O.A., 1986, A computational method for ordinary differential equations and error estimation. MSc dissertation, University of Ilorin, Nigeria (unpublished); Taiwo, O.A., 2002, Exponential fitting for the solution of two point boundary value problem with cubic spline collocation tau-method. International Journal of Computer Mathematics, 79(3), 229–306.] are discussed and applied to some second order non-linear problems. They are standard collocation, perturbed collocation, and exponentially fitted collocation. Numerical examples are given to illustrate the accuracy, efficiency and computational cost.     

Error analysis of the ultraspherical spectral-collocation method for high-index IAEs

The subject of integral-algebraic equations (IAEs) with index?>?2 is new in the literature and there are a few available results which investigate these systems. We will deal here with a particular class of IAEs of index-3 and analyse ultraspherical polynomials collocation method as well as convergence analysis for the numerical solution of these systems. Our error analysis of collocation method is based on polynomial approximation theory and results related to the interpolation error in Sobolev space. Numerical examples illustrate the theoretical results.     

Continuous Volterra-Runge-Kutta methods for integral equations with pure delay

In the following we give an analysis of the local superconvergence properties of piecewise polynomial collocation methods and related continuous Runge-Kutta-type methods for Volterra integral equations with constant delay. We show in particular that (in contrast to delay differential equations) collocation at the Gauss points does not lead to higher-order convergence and thusm-stage Gauss-Runge-Kutta methods for delay Volterra equations do not possess the orderp=2m.     

Legendre–Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations

We propose new Legendre–Gauss collocation algorithms for ordinary differential equations. We also design Legendre–Gauss-type collocation algorithms for time-dependent nonlinear partial differential equations. The suggested algorithms enjoy spectral accuracy in both time and space, and can be implemented in a fast and stable manner. Numerical results exhibit the effectiveness.     

Chebyshev pseudospectral method for wave equation with absorbing boundary conditions that does not use a first order hyperbolic system

The analysis and solution of wave equations with absorbing boundary conditions by using a related first order hyperbolic system has become increasingly popular in recent years. At variance with several methods which rely on this transformation, we propose an alternative method in which such hyperbolic system is not used. The method consists of approximation of spatial derivatives by the Chebyshev pseudospectral collocation method coupled with integration in time by the Runge-Kutta method. Stability limits on the timestep for arbitrary speed are calculated and verified numerically. Furthermore, theoretical properties of two methods by Jackiewicz and Renaut are derived, including, in particular, a result that corrects some conclusions of these authors. Numerical results that verify the theory and illustrate the effectiveness of the proposed approach are reported.     

A symmetric variational finite element-boundary integral equation coupling method

This paper is concerned with the development of a mixed variational principle for coupling finite element and boundary integral methods in interface problems, using the generalized Poisson's equation as a prototype situation. One of its primary objectives is to compare the performance of fully variational procedures with methods that use collocation for the treatment of boundary integral equations. A distinctive feature of the new variational principle is that the discretized algebraic equations for the coupled problem are automatically symmetric since they are all derived from a single functional. In addition, the condition that the flux remain continuous across interfaces is satisfied naturally. In discretizing the problem within inhomogeneous or loaded regions, domain finite elements are used to approximate the field variable. On the other hand, only boundary elements are used for regions where the medium is homogeneous and free of external agents. The corresponding integral equations are discretized both by fully variational and by collocation techniques. Results of numerical experiments indicate that the accuracy of the fully variational procedure is significantly greater than that of collocation for the complete interface problem, especially for complex disturbances, at little additional computational cost. This suggests that fully variational procedures may be preferable to collocation, not only in dealing with interface problems, but even for solving integral equations by themselves.