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1.
Michael Hanke 《Numerische Mathematik》1988,54(1):79-90
Summary The aim of this note is to extend some results on least-squares collocation methods and to prove the convergence of a least-squares collocation method applied to linear differential-algebraic equations. Some numerical examples are presented. 相似文献
2.
Wilhelm Heinrichs 《Numerical Algorithms》2007,44(1):1-10
A least-squares spectral collocation method for the one-dimensional inviscid Burgers equation is proposed. This model problem
shows the stability and high accuracy of these schemes for nonlinear hyperbolic scalar equations. Here we make use of a least-squares
spectral approach which was already used in an earlier paper for discontinuous and singular perturbation problems (Heinrichs,
J. Comput. Appl. Math. 157:329–345, 2003). The domain is decomposed in subintervals where continuity is enforced at the interfaces. Equal order polynomials are used
on all subdomains. For the spectral collocation scheme Chebyshev polynomials are employed which allow the efficient implementation
with Fast Fourier Transforms (FFTs). The collocation conditions and the interface conditions lead to an overdetermined system
which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear
systems. The scheme exhibits exponential convergence where the exact solution is smooth. In parts of the domain where the
solution contains discontinuities (shocks) the spectral solution displays a Gibbs-like behavior. Here this is overcome by
some suitable exponential filtering at each time level. Here we observe that by over-collocation the results remain stable
also for increasing filter parameters and also without filtering. Furthermore by an adaptive grid refinement we were able
to locate the precise position of the discontinuity. Numerical simulations confirm the high accuracy of our spectral least-squares
scheme.
相似文献
3.
In this paper, we investigate the application of radial basis functions (RBFs) for the approximation with collocation of the Stokes problem. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions with decreasing scaling factors. We use symmetric collocation and give sufficient conditions for convergence and consider stability analysis. Numerical experiments support the theoretical results. 相似文献
4.
We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied. 相似文献
5.
Hsin‐Yun Hu Jiun‐Shyan Chen Wei Hu 《Numerical Methods for Partial Differential Equations》2011,27(3):554-580
Solving partial differential equations (PDE) with strong form collocation and nonlocal approximation functions such as orthogonal polynomials, trigonometric functions, and radial basis functions exhibits exponential convergence rates; however, it yields a full matrix and suffers from ill conditioning. In this work, we discuss a reproducing kernel collocation method, where the reproducing kernel (RK) shape functions with compact support are used as approximation functions. This approach offers algebraic convergence rate, but the method is stable like the finite element method. We provide mathematical results consisting of the optimal error estimation, upper bound of condition number, and the desirable relationship between the number of nodal points and the number of collocation points. We show that using RK shape function for collocation of strong form, the degree of polynomial basis functions has to be larger than one for convergence, which is different from the condition for weak formulation. Numerical results are also presented to validate the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 554–580, 2011 相似文献
6.
《Mathematical Methods in the Applied Sciences》2018,41(13):4986-5002
In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate. 相似文献
7.
Summary.
We propose an approximation method for
periodic pseudodifferential equations,
which yields higher convergence rates in
Sobolev spaces with negative order
than the collocation method. The main
idea consists in correcting the usual
collocation solution in a certain way
by the solution of a small Galerkin
system for the same equation. Both
trigonometric and spline approximation
methods are considered. In most of
the cases our convergence result
even improves that of the qualocation method.
Received
January 3, 1994 / Revised version received August 17,
1994 相似文献
8.
Johannes Elschner 《Mathematische Nachrichten》1988,139(1):309-319
This paper analyses the convergence of spline approximation methods for strongly elliptic singular integral equations on a finite interval. We consider collocation by smooth polynomial splines of odd degree multiplied by a weight function and a Galerkin-Petrov method with spline trial functions of even degree and piecewise constant test functions. We prove the stability of the methods in weighted Sobolev spaces and obtain the optimal orders of convergence in the case of graded meshes. 相似文献
9.
Collocation approximations with harmonic basis functions tothe solution of the harmonic Dirichlet problem are investigated.The choice of collocation points for a best local approximationis discussed, and a result is given in terms of the abscissaeof some best quadrature formulae. A global near-best approximationis obtained by adding a correction term to the collocation approximation,utilizing basic properties of the Green's function. Numericalexamples are given, demonstrating the great improvement achieved.The same correction term can also improve on least-squares approximationsand Galerkin approximations, and the results can easily be adaptedto deal with mixed harmonic boundary value problems. 相似文献
10.
S.W. Chi J.S. Chen H. Luo H.Y. Hu L. Wang 《Numerical Methods for Partial Differential Equations》2013,29(3):818-842
Strong form collocation with radial basis approximation, called the radial basis collocation method (RBCM), is introduced for the numerical solution of elastodynamics. In this work, the proper weights for the boundary collocation equations to achieve the optimal convergence in elastodynamics are first derived. The von Neumann method is then introduced to investigate the dispersion characteristics of the semidiscrete RBCM equation. Very small dispersion error (< 1%) in RBCM can be achieved compared to linear and quadratic finite elements. The stability conditions of the RBCM spatial discretization in conjunction with the central difference temporal discretization are also derived. We show that the shape parameter of the radial basis functions not only has strong influence on the dispersion errors, it also has profound influence on temporal stability conditions in the case of lumped mass. Further, our stability analysis shows that, in general, a larger critical time step can be used in RBCM with central difference temporal discretization than that for finite elements with the same temporal discretization. Our analysis also suggests that although RBCM with lumped mass allows a much larger critical time step than that of RBCM with consistent mass, the later offers considerably better accuracy and should be considered in the transient analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 相似文献
11.
The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation. 相似文献
12.
Yiorgos-Sokratis Smyrlis 《BIT Numerical Mathematics》2006,46(1):163-194
We investigate the Method of Fundamental Solutions (MFS) for the solution of certain elliptic boundary value problems. In
particular, we study the case in which the number of collocation points exceeds the number of singularities, which leads to
an over-determined linear system. In such a case, the resulting linear system is over-determined and the proposed algorithm
chooses the approximate solution for which the error, when restricted to the boundary, minimizes a suitably defined discrete
Sobolev norm. This is equivalent to a weighted least-squares treatment of the resulting over-determined system. We prove convergence
of the method in the case of the Laplace’s equation with Dirichlet boundary data in the disk. We develop an alternative way
of implementing the numerical algorithm, which avoids the inherent ill-conditioning of the MFS matrices. Finally, we present
numerical experiments suggesting that introduction of Sobolev weights improves the approximation.
AMS subject classification (2000) 35E05, 35J25, 65N12, 65N15, 65N35, 65T50 相似文献
13.
Meshless method with ridge basis functions 总被引:1,自引:0,他引:1
Meshless collocation methods for the numerical solutions of PDEs are increasingly adopted due to their advantages including efficiency and flexibility, and radial basis functions are popularly employed to represent the solutions of PDEs. Motivated by the advantages of ridge basis function representation of a given function, such as the connection to neural network, fast convergence as the number of terms is increased, better approximation effects and various applications in engineering problems, a meshless method is developed based on the collocation method and ridge basis function interpolation. This method is a truly meshless technique without mesh discretization: it neither needs the computation of integrals, nor requires a partition of the region and its boundary. Moreover, the method is applied to elliptic equations to examine its appropriateness, numerical results are compared to that obtained from other (meshless) methods, and influence factors of accuracy for numerical solutions are analyzed. 相似文献
14.
Magda Rebelo 《Journal of Computational and Applied Mathematics》2010,234(9):2859-2869
This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method. 相似文献
15.
A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions 下载免费PDF全文
A. H. Bhrawy E. H. Doha D. Baleanu R. M. Hafez 《Mathematical Methods in the Applied Sciences》2015,38(14):3022-3032
In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
16.
This paper is concerned with the numerical solution using orthogonalcollocation of a nonlinear integro-differential equation whichexhibits certain features arising in the modelling of catalyticcombustion in a cylindrical reactor taking account of radiation.These features imply that the available convergence theory oforthogonal collocation cannot be applied directly. Optimal convergenceresults of the orthogonal collocation approximation are provedfor our equation, and these are illustrated by numerical experiments.In addition superconvergence results for the iterated collocationsolution are obtained. 相似文献
17.
A backward euler orthogonal spline collocation method for the time‐fractional Fokker–Planck equation 下载免费PDF全文
Graeme Fairweather Haixiang Zhang Xuehua Yang Da Xu 《Numerical Methods for Partial Differential Equations》2015,31(5):1534-1550
We formulate and analyze a novel numerical method for solving a time‐fractional Fokker–Planck equation which models an anomalous subdiffusion process. In this method, orthogonal spline collocation is used for the spatial discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1534–1550, 2015 相似文献
18.
The spline collocation method for a class of biperiodic strongly elliptic pseudodifferential operators is considered. As trial
functions tensor products of odd degree splines are used and the collocation is imposed at the nodal points of the tensor
product mesh. It is shown that the collocation problem is uniquely solvable if the maximum mesh length is small enough. Moreover,
the approximation is stable and quasioptimal with respect to a norm depending on the order of the operator and the degree
of approximating splines. Some convergence results are given for general and quasiuniform meshes. The results cover for example
the single layer and the hypersingular operators. 相似文献
19.
Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically
stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation
methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation
algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution
are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require
mesh-ratio restrictions. Numerical results that support the theoretical findings are also given.
Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001 相似文献
20.
A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary
conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into
a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived
for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The
fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers.
The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline
collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method. 相似文献