Collocation Methods for Weakly Singular Second-kind Volterra Integral Equations with Non-smooth Solution

Collocation type methods are studied for the numerical solutionof the weakly singular Volterra integral equation of the secondkind: where the solution (t) is assumedto have the form f(t) = x(t)+r^?(t), x and being sufficientlysmooth. The solution is approximated near zero by a linear combinationof powers of t?, and away from zero by the usual polynomialrepresentation. Convergence is proved and many numerical experimentsare carried out with examples from the literature. A comparisonis made with a method of Brunner & Norsett (1981), originallydeveloped for (1) with a smooth solution. Special attentionis paid to the numerical approximation of the so-called momentintegrals which emerge in the collocation scheme.     

Superconvergence of a Collocation-type Method for Hummerstein Equations

This paper considers the numerical solution of Hammerstein equationsof the form by a collocation method applied not to this equation, but ratherto an equivalent equation for z(t) :=g(t, y(t)). The desiredapproximation to y is then obtained by use of the (exact) equation In an earlier paper, questions of existence and optimal convergenceof the respective approximations to z and y were examined. Inthis sequel, collocation approximations to z are sought in certainpiecewise polynomial function spaces, and analogous of knownsuperconvergence results for the iterated collocation solutionof (linear) second-kind Fredhoim integral equations are statedand proved for the approximation to y.     

On the Effect of Boundary Conditions in Collocation by Polynomial Splines for the Solution of Boundary Value Problems in Ordinary Differential Equations

Consider the numerical solution of a boundary-value problemfor a differential equation of order m using collocation ofa polynomial spline of degree n m on a uniform mesh of sizeh. We describe several collocation schemes which differ onlyin the boundary collocation conditions and which include a "natural"spline collocation scheme. Taking account of derived asymptoticerror bounds most of which are, roughly speaking, of O(hn^12m+1), we discuss the computational effectiveness of the variousschemes.     

Sinc-Collection Methods for Two-Point Boundary Value Problems

A collocation method is developed for the approximate solutionof two-point boundary-value problems with mixed boundary conditions.The method is based on replacing the exact solution by a linearcombination of Sinc functions. No integrals need to be evaluatedapproximately when setting up the resulting system of linearequations. The error of the method converges to zero like O(exp(-cN²)),as N, where N is the number of collocation points used, andwhere c is a positive constant independent of N. It is claimedthat the method is superior to the Sinc-Galerkin method dueto its simple implementation and possible extensions to moregeneral boundary-value problems.     

On the Sloan iteration applied to integral equations of the first kind

When the piecewise constant collocation method is used to solvean integral equation of the first kind with logarithmic kernel,the convergence rate is O(h) in the L₂ norm. In this note weshow that O(h³) or O(h⁵) convergence in any Sobolev norm (andthus, for example, in L) may be obtained by a simple cheap postprocessingof the original collocation solution. The construction of thepostprocessor is based on writing the first kind equation asa second kind equation, and applying the Sloan iteration tothe latter equation. The theoretical convergence rates are verifiedin a numerical example.     

The numerical solution of a non-linear boundary integral equation on smooth surfaces

We study a boundary integral equation method for solving Laplace'sequation u=0 with non-linear boundary conditions. This non-linearboundary value problem is reformulated as a non-linear boundaryintegral equation, with u on the boundary as the solution beingsought. The integral equation is solved numerically by usingthe collocation method, with piecewise quadratic functions usedas approximations to u. Convergence results are given for thecases where (1) the original surface is used, and (2) the surfaceis approximated by piecewise quadratic interpolation. In addition,we define and analyze a two-grid iteration method for solvingthe non-linear system that arises from the discretization ofthe boundary integral equation. Numerical examples are given;and the paper concludes with a short discussion of the relativecost of different parts of the method. This work was supported in part by NSF grant DMS-9003287.     

On the stability of Runge-Kutta methods for delay integral equations

Summary We present a class of Runge-Kutta methods for the numerical solution of a class of delay integral equations (DIEs) described by two different kernels and with a fixed delay . The stability properties of these methods are investigated with respect to a test equation with linear kernels depending on complex parameters. The results are then applied to collocation methods. In particular we obtain that any collocation method for DIEs, resulting from anA-stable collocation method for ODEs, with a stepsize which is submultiple of the delay , preserves the asymptotic stability properties of the analytic solutions.This work was supported by CNR (Italian National Council of Research)     

Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) \(q^{\prime \prime }(t)+Mq(t)=f(q(t))\) with a principal frequency matrix \(M\in \mathbb {R}^{d\times d}\). If \(M\) is symmetric and positive semi-definite and \(f(q) = -\nabla U(q)\) for a smooth function \(U(q)\), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian \(H(q,p)=p^{T}p/2+q^{T}Mq/2+U(q),\) where \(p = q'\). The solution of this system is a nonlinear multi-frequency oscillator. The new trigonometric Fourier collocation method takes advantage of the special structure brought by the linear term \(Mq\), and its construction incorporates the idea of collocation methods, the variation-of-constants formula and the local Fourier expansion of the system. The properties of the new methods are analysed. The analysis in the paper demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary order and when \(M\rightarrow 0\), each trigonometric Fourier collocation method creates a particular Runge–Kutta–Nyström-type Fourier collocation method, which is symplectic under some conditions. This allows us to obtain arbitrary high-order symplectic methods to deal with a special and important class of systems of second-order ODEs in an efficient way. The results of numerical experiments are quite promising and show that the trigonometric Fourier collocation methods are significantly more efficient in comparison with alternative approaches that have previously appeared in the literature.     

On a Functional Differential Equation

This paper considers some analytical and numerical aspects ofthe problem defined by an equation or systems of equations ofthe type (d/dt)y(t) = ay(t)+by(t), with a given initial conditiony(0) = 1. Series, integral representations and asymptotic expansions fory are obtained and discussed for various ranges of the parametersa, b and (> 0), and for all positive values of the argumentt. A perturbation solution is constructed for |1–| <<1, and confirmed by direct computation. For > 1 the solutionis not unique, and an analysis is included of the eigensolutionsfor which y(0) = 0. Two numerical methods are analysed and illustrated. The first,using finite differences, is applicable for < 1, and twotechniques are demonstrated for accelerating the convergenceof the finite-difference solution towards the true solution.The second, an adaptation of the Lanczos method, is applicablefor any > 0, though an error analysis is available onlyfor < 1. Numerical evidence suggests that for > 1 themethod still gives good approximations to some solution of theproblem.     

Asymptotic Properties of Collocation Matrix Norms 2: Piecewise Polynomial Approximation

This paper considers a matrix related to the solution by piecewisepolynomial collocation using n subintervals of an mth-orderordinary differential boundary-value problem. It is shown thatif the maximum subinterval size tends to zero as the matrix norm tends to the norm of an operator related tothe differential equation, under the assumption that the collocationpoints in each subinterval are assumed to be distributed identicallyand their associated interpolatory quadrature weights are positive.     

Global blow-up of separable solutions of the vorticity equation

In this paper we construct solutions to the equation on a finite interval in y which blow-up globallyin finite time. This equation arises in a number of physicalsituations and can be derived from the vorticity equation bylooking for stagnation-point type separable solutions for thetwo-dimensional streamfunction of the form xu(y, t). In theparticular application which has prompted the investigationreported in this paper, (*) is solved subject to boundary conditionsinvolving ²u/y². For this type of boundary condition the phenomenonof blow-up was first observed numerically by solving the initial-boundary-valueproblem for (*). These computations reveal that, depending onthe parameter combinations chosen, the solution to the initial-valueproblem may either blow-up globally in finite time or approacha steady state as t . Using the computations as a guide weconstruct the analytic behaviour of the solution close to theblow-up time using the methods of formal asymptotics.     

A Posteriori Error Bounds for Linear Elliptic Equations of Potential Type

Second-order Linear elliptic partial differential equations of potential type with Dirichlet (type 1) or Neumann (type II)boundary conditions on a simply-connected two-dimensional domainare considered. Conjugate problems, that is, a pair of one type1 and one type II problem, are introduced along with an auxiliaryeppiptic system of two equations in such a way that the energiesof the given problem, its conjugate problem, and the auxiliarysystem add to a known constant. There result two-sided boundsfor the energy of the given problem and, as a consequence, aposteriori error bounds for the norm of the difference of anapproximate solution and the exact solution of the problem.A method by which the amount of computation required to obtainthe a posteriori error bounds can be almost halved in many casesof practical interest is given. A posteriori error bounds forapproximate solutions of the auxiliary system are also given.     

Bivariational Methods for Hammerstein Integral Equations

Bivariational methods are presented for nonlinear integral equationsof Ham-merstein type = Kf(). With appropriate conditions onK and f, various upper and lower bounding functionals are derivedfor inner products and associated with the solution . Inthe latter case, suitable choices for g lead to point wise boundson both and its derivative. The methods are tested on a pendulumequation, and encouraging accuracy is obtained using simpletrial vectors.     

Vibration of an infinite membrane supported by an array of posts

The vibration of an infinite membrane supported by a squareor triangular array of circular posts is studied by eigenfunctionexpansion and collocation. The fundamental frequencies are foundto be highly sensitive to both small and large relative postradii b. The singular nature of the frequency as b 0 is expressedin an asymptotic formula.     

On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization

^** Email: bause{at}am.uni-erlangen.de Due to the increasing use of higher-order methods in computationalfluid dynamics, the question of optimal approximability of theNavier–Stokes equations under realistic assumptions onthe data has become important. It is well known that the regularitycustomarily hypothesized in the error analysis for parabolicproblems cannot be assumed for the Navier–Stokes equations,as it depends on non-local compatibility conditions for thedata at time t = 0, which cannot be verified in practice. Takinginto account this loss of regularity at t = 0, improved convergenceof the order (min{h^(5/2)–,h³/t^(1/4)+}), for any >0, is shown locally in time for the spatial discretization ofthe velocity field by (non-)conforming finite elements of third-orderapproximability properties. The error estimate itself is provedby energy methods, but it is based on sharp a priori estimatesfor the Navier–Stokes solution in fractional-order spacesthat are derived by semigroup methods and complex interpolationtheory and reflect the optimal regularity of the solution ast 0.     

A Fast Galerkin Algorithm for Singular Integral Equations

A recent paper (Delves, 1977) described a variant of the Galerkinmethod for linear Fredholm integral equations of the secondkind with smooth kernels, for which the total solution timeusing N expansion functions is (N² ln N) compared with the standardGalerkin count of (N³). We describe here a modification of thismethod which retains this operations count and which is applicableto weakly singular Fredholm equations of the form where K₀(x, y) is a smooth kernel and Q contains a known singularity.Particular cases treated in detail include Fredholm equationswith Green's function kernels, or with kernels having logarithmicsingularities; and linear Volterra equations with either regularkernels or of Abel type. The case when g(x) and/or f(x) containsa known singularity is also treated. The method described yieldsboth a priori and a posteriori error estimates which are cheapto compute; for smooth kernels (Q = 1) it yields a modifiedform of the algorithm described in Delves (1977) with the advantagethat the iterative scheme required to solve the equations in(N²) operations is rather simpler than that given there.     

Collocation Methods and Their Modifications for Cauchy Singular Integral Equations on the Interval

In this paper we present polynomial collocation methods and their modi.cations for the numerical solution of Cauchy singular integral equations over the interval [-1, 1]. More precisely, the operators of the integral equations have the form with piecewise continuous coefficients a and b, and with a Jacobi weight . Using the splitting property of the singular values of the collocation methods, we obtain enough stable approximate methods to .nd the least square solution of our integral equation. Moreover, the modifications of the collocation methods enable us to compute kernel and cokernel dimensions of operators from a C*-algebra, which is generated by operators of the Cauchy singular integral equations.     

Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations

Summary We discuss the application of a class of spline collocation methods to first-order Volterra integro-differential equations (VIDEs) which contain a weakly singular kernel (t–s)^– with 0<<1. It will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes. The grading exponents of graded meshes used are not greater thanm (the polynomial degree) which is independent of . This is in contrast to the theories of spline collocation methods for Volterra (or Fredholm) integral equation of the second kind. Numerical examples are given to illustrate the theoretical results.     

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods appliedto boundary value problems for the positive definite Helmholtz-typeproblem –U + ²U = 0 in a bounded or unbounded domain,with the parameter real and possibly large. Applications arisein the implementation of space–time boundary integralmethods for the heat equation, where is proportional to 1/(t),and t is the time step. The corresponding layer potentials arisingfrom this problem depend nonlinearly on the parameter and havekernels which become highly peaked as , causing standard discretizationschemes to fail. We propose a new collocation method with arobust convergence rate as . Numerical experiments on a modelproblem verify the theoretical results.     

An Adaptive Greedy Algorithm for Solving Large RBF Collocation Problems

The solution of operator equations with radial basis functions by collocation in scattered points leads to large linear systems which often are nonsparse and ill-conditioned. But one can try to use only a subset of the data for the actual collocation, leaving the rest of the data points for error checking. This amounts to finding sparse approximate solutions of general linear systems arising from collocation. This contribution proposes an adaptive greedy method with proven (but slow) linear convergence to the full solution of the collocation equations. The collocation matrix need not be stored, and the progress of the method can be controlled by a variety of parameters. Some numerical examples are given.