An iterative method based upon Padé approximants

Damil, N.; Potier‐Ferry, Michel; Najah, A.; Chari, Rachid; Lahmam, Hassane

doi:10.1002/(sici)1099-0887(199910)15:10<701::aid-cnm283>3.0.co;2-l

Cited by 37 publications

(24 citation statements)

References 8 publications

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“…We can accelerate the convergence of the homotopyperturbation technique presented in the introduction by using another homotopy transformation introduced in an earlier work [9,15] and in another framework. To solve the initial value problem (5), we propose, in this work, a high order implicit algorithm which combines: a time discretization, a homotopy transformation, a perturbation technique and a space discretization.…”

Section: The Proposed Algorithmmentioning

confidence: 99%

“…There is no general rule to derive a homotopy transformation as (9). The latter one is exactly the same as proposed in [15] in a static framework after several attempts.…”

Section: The Proposed Algorithmmentioning

confidence: 99%

“…The idea is to apply a homotopy transformation and to compute series with respect to the homotopy parameter, see [1,8]. We refer to recent papers [9,15] for a general discussion and an optimisation of these high order iterative algorithms. A key point is the possibility to choose the operator that will be triangulated and to compute many time steps with a single matrix triangulation.…”

Section: Introductionmentioning

confidence: 99%

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A high order implicit algorithm for solving instationary non-linear problems

Jamal

Braikat

Boutmir³

et al. 2002

Computational Mechanics

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In this paper a high order implicit algorithm is developed for solving instationary non-linear problems. This generic numerical method combines four mathematical techniques: a time discretization, a homotopy transformation, a perturbation technique and a space discretization. The time integration is performed by classical implicit schemes (Euler implicit for problems with a first order time derivative and Newmark for second order). The timediscretization leads to non-linear equations. In this paper a new technique is proposed to solve iteratively the latter equations. The key points in this approach are, first a high order solver based on perturbation techniques, second the possibility of choosing the iteration operator, which limits the number of matrices to be triangulated. To illustrate the performance of the proposed algorithm two examples are considered: the Korteweg-de Vries equation (KdV) and the non-linear oscillations of a 2D elastic pendulum. IntroductionDuring the last few years, a considerable research amount has been done in order to elaborate efficient algorithms for solving non-linear evolution problems. A variety of numerical methods for these time dependent problems has been proposed. The time discretization of a non-linear evolution equation leads to a non-linear equation to define the solution at the next step. The most common methods to solve the latter equation are the classical iterative algorithms for evolution problems [10, 17, 18] and predictor-corrector ones for dynamic structural problems [2,3,7,12,19]. These algorithms permit ones to get the solution of the considered non-linear time dependent problems in a step wise manner and a full point by point description is then obtained. These algorithms work very well but they may be costly in term of computation time.In this paper, we propose to use high order iteration techniques to solve the non-linear problems deduced from time discretization. These methods are well established and efficient to solve path-following problems and they can be applied to a lot of physical models, see for instance [4,6,16]. It is also possible to solve in that way non-linear problems, that do not depend on a scalar parameter. The idea is to apply a homotopy transformation and to compute series with respect to the homotopy parameter, see [1,8]. We refer to recent papers [9, 15] for a general discussion and an optimisation of these high order iterative algorithms. A key point is the possibility to choose the operator that will be triangulated and to compute many time steps with a single matrix triangulation. An algorithm of this type has been proposed recently in [5,14]. A variant of the latter method will be presented and tested here, that leads to a strong reduction of the number of matrix inversions.To illustrate the proposed algorithm, let us consider the following non-linear evolution problem:where f ð:Þ is a spatial differential operator, u is the unknown and u 0 is the initial condition. Using, for instance, an Euler implicit scheme, the time discretizatio...

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Section: The Proposed Algorithmmentioning

confidence: 99%

“…There is no general rule to derive a homotopy transformation as (9). The latter one is exactly the same as proposed in [15] in a static framework after several attempts.…”

Section: The Proposed Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A high order implicit algorithm for solving instationary non-linear problems

Jamal

Braikat

Boutmir³

et al. 2002

Computational Mechanics

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“…Eventually, many analytical and numerical methods have been established and used to reach this goal. Some of the methods are; the perturbation method [1][2][3][4][5], the homotopy perturbation method [3][4][5][6][7], the delta perturbative method [8], the Modified Decomposition method [9][10][11][12][13][14][15][16], the Adomian Decomposition method [15][16][17][18][19][20][21][22], the Laplace Decomposition method [15,16,[22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

“…When f (u) x = uu x with α = 0 and β = 1, equation (1) turns into the alternative regularized long-wase equation proposed by Peregrine [29] and Benjamin [30]. This equation characterizes a balance between nonlinear and dispersive effects, yet it does not take place in calculation of dissipation.…”

Section: Introductionmentioning

confidence: 99%

Applications of the decomposition methods to some nonlinear partial differential equations

Gündoğdu¹,

Gözükızıl²

2018

ntmsci

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Abstract:In this paper, we consider the inhomogeneous Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation with its initial condition. The decomposition methods such as Adomian and Laplace are applied to this equation. Then, the solution satisfying the given initial condition is gained by using each method. To demonstrate these decomposition methods are how effective and convenient for solving the nonlinear partial differential equations(NLPDEs), solutions of some NLPDEs subjected to initial conditions are attained by these method. In the application of the method, it is clearly noticed that there is no need to convert the nonlinear terms into the linear ones due to the Adomian polynomials. In addition to this simplicity, it is seen that each method gives the same solution with a fast convergence by taking the advantage of the noise terms.

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A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

Cadou

Potier‐Ferry²

2009

Numer Methods Biomed Eng

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SUMMARYAn iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non-linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub-steps. In the first sub-step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub-step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well-known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non-linear solvers with few matrix factorizations and few iterations.

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An iterative method based upon Padé approximants

Abstract: SUMMARYWe present a new iterative method based on PadeÂ approximants for numerical analysis of non-linear problems. It improves on the classical iterative Newton methods.

Cited by 37 publications

References 8 publications

A high order implicit algorithm for solving instationary non-linear problems

A high order implicit algorithm for solving instationary non-linear problems

Applications of the decomposition methods to some nonlinear partial differential equations

A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

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