共查询到20条相似文献,搜索用时 15 毫秒
1.
Bernard Bialecki 《Numerische Mathematik》1991,59(1):413-429
Summary An Alternating Direction Implicit method is analyzed for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to some separable, second order, linear, elliptic partial differential equations in rectangles. On anNxN partition, with Jordan's selection of the acceleration parameters, the method requiresO(N
2 ln
2
N) arithmetic operations to produce an approximation whose accuracy, in theH
1-norm, is that of the collocation solution. 相似文献
2.
We use the generalized L1 approximation for the Caputo fractional derivative, the second-order fractional quadrature rule approximation for the integral term, and a classical Crank-Nicolson alternating direction implicit (ADI) scheme for the time discretization of a new two-dimensional (2D) fractional integro-differential equation, in combination with a space discretization by an arbitrary-order orthogonal spline collocation (OSC) method. The stability of a Crank-Nicolson ADI OSC scheme is rigourously established, and error estimate is also derived. Finally, some numerical tests are given. 相似文献
3.
S. V. Gaidomak 《Computational Mathematics and Mathematical Physics》2008,48(7):1161-1180
Some classes of singular systems of partial differential equations with variable matrix coefficients and internal hyperbolic structure are considered. The spline collocation method is used to numerically solve such systems. Sufficient conditions for the convergence of the numerical procedure are obtained. Numerical results are presented. 相似文献
4.
5.
S. Arul Veda Manickam Amiya K. Pani Sang K. Chung 《Numerical Methods for Partial Differential Equations》1998,14(6):695-716
A second-order splitting method is applied to a KdV-like Rosenau equation in one space variable. Then an orthogonal cubic spline collocation procedure is employed to approximate the resulting system. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index 1. Error estimates in L2 and L∞ norms have been obtained for the semidiscrete approximations. For the temporal discretization, the time integrator RADAU5 is used for the resulting system. Some numerical experiments have been conducted to validate the theoretical results and to confirm the qualitative behaviors of the Rosenau equation. Finally, orthogonal cubic spline collocation method is directly applied to BBM (Benjamin–Bona–Mahony) and BBMB (Benjamin–Bona–Mahony–Burgers) equations and the well-known decay estimates are demonstrated for the computed solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 695–716, 1998 相似文献
6.
7.
Bernard Bialecki 《Numerical Methods for Partial Differential Equations》1999,15(3):285-303
Superconvergence phenomena have been observed numerically in the piecewise Hermite bicubic orthogonal spline collocation solution of Poisson's equation on a rectangle. The purpose of this article is to demonstrate theoretically the superconvergent fourth‐order accuracy in the first‐order partial derivatives of the collocation solution at the partition nodes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 285–303, 1999 相似文献
8.
Mehdi Dehghan Akbar Mohebbi 《Numerical Methods for Partial Differential Equations》2009,25(1):232-243
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 相似文献
9.
Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonal. We construct an orthogonal basis for the n-order(n ≥ 3) algebraic-trigonometric spline space in order to resolve the theoretical problem that there is not an explicit orthogonal basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions,which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions. 相似文献
10.
《Numerical Methods for Partial Differential Equations》2018,34(2):555-574
The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two‐dimensional multiterm fractional convection‐diffusion reaction equation with variable coefficients. The optimal estimates in Hj (j = 0, 1, 2) norms at each time step are derived. Also, estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm. 相似文献
11.
交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性. 相似文献
12.
We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k2 + h2) for the difference solution of linear hyperbolic equation utt + 2αut + β2u = uxx + uyy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001 相似文献
13.
Pradip Roul Kiran Thula VMK Prasad Goura 《Mathematical Methods in the Applied Sciences》2019,42(8):2613-2630
This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM. 相似文献
14.
A backward euler orthogonal spline collocation method for the time‐fractional Fokker–Planck equation
下载免费PDF全文
![点击此处可从《Numerical Methods for Partial Differential Equations》网站下载免费的PDF全文](/ch/ext_images/free.gif)
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
15.
Bernard Bialecki 《Numerical Algorithms》1994,8(2):167-184
Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N
2log2
N) andO(N
2log2log2
N) arithmetic operations, respectively. 相似文献
16.
Ryan I. Fernandes 《Numerische Mathematik》1997,77(2):223-241
Summary. Piecewise Hermite bicubic orthogonal spline collocation Laplace-modified and alternating-direction schemes for the approximate
solution of linear second order hyperbolic problems on rectangles are analyzed. The schemes are shown to be unconditionally
stable and of optimal order accuracy in the and discrete maximum norms for space and time, respectively. Implementations of the schemes are discussed and numerical results
presented which demonstrate the accuracy and rate of convergence using various norms.
Received November 7, 1994 / Revised version received April 29, 1996 相似文献
17.
Muhammad Asif Nadeem Haider Qasem Al-Mdallal Imran Khan 《Numerical Methods for Partial Differential Equations》2020,36(6):1962-1981
We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one- and two-dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one- and two-dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate. 相似文献
18.
Zhiyue Zhang Dingwen Deng 《Numerical Methods for Partial Differential Equations》2007,23(6):1530-1559
A modified backward difference time discretization is presented for Galerkin approximations for nonlinear hyperbolic equation in two space variables. This procedure uses a local approximation of the coefficients based on patches of finite elements with these procedures, a multidimensional problem can be solved as a series of one‐dimensional problems. Optimal order H01 and L2 error estimates are derived. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
19.
Suruchi Singh Swarn Singh 《Numerical Methods for Partial Differential Equations》2020,36(5):1028-1043
A high order modified nodal bi-cubic spline collocation method is proposed for numerical solution of second-order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency. 相似文献
20.
Shanshan Wang Luming Zhang Ran Fan 《Journal of Computational and Applied Mathematics》2011,235(8):1993-2005
In this paper, discrete-time orthogonal spline collocation schemes are proposed for the nonlinear Schrödinger equation with wave operator. These schemes are constructed by using orthogonal spline collocation approaches combined with finite difference methods. The conservative property, the convergence, and the stability of these methods are theoretically analyzed and also verified by extensive numerical experiments. In addition, some interesting phenomena which require further theoretical analysis are discussed numerically. 相似文献