Efficiency of Methods for Second-Order Problems

Gladwell, I.; Thomas, R. M.

doi:10.1093/imanum/10.2.181

Cited by 15 publications

(6 citation statements)

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“…Peat and Thomas [20], after extensive numerical experiments, concluded that the schemes proposed by Cooper and Butcher are, in general, the most efficient schemes for integration of stiff problems. Gladwell and Thomas [16] recommended this scheme for the two-stage Gauss method. Cooper and Vignesvaran [10] proposed a scheme which is a generalization of the basic scheme (6).…”

Section: Motivationmentioning

confidence: 99%

Improving Rate of Convergence of an Iterative Scheme With Extra Sub-Steps for Two Stage Gauss Method

Vigneswaran¹,

Kajanthan²

2017

Int. J. of Pure and Appl. Math.

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The non-linear equations, when implementing implicit Runge-Kutta methods, may be solved by a modified Newton scheme and by several linear iteration schemes which sacrificed superlinear convergence for reduced linear algebra costs. A linear scheme of this type was proposed, which requires some additional computation in each iteration step. The rate of convergence of this scheme is examined when it is applied to the scalar test problem

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Section: Motivationmentioning

confidence: 99%

Improving Rate of Convergence of an Iterative Scheme With Extra Sub-Steps for Two Stage Gauss Method

Vigneswaran¹,

Kajanthan²

2017

Int. J. of Pure and Appl. Math.

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“…Though, Gladwell and Thomas [2], Murugesan and Balasubramanian [4], Van Daele et al, [8], Vander Houmen and Sommeijer [9] have contributed for the determination of discrete solutions for the second order IVPs, it is observed, interestingly, that very little work has been carried out in getting approximate solutions for the system of second order IVPs involving multivariables of the form…”

Section: Second Order Multivariable State-space and Singular Sysmentioning

confidence: 99%

Numerical Analysis of second order Multivariable Linear Systems using He’s Variational Iteration Method

Sekar¹,

Ravikumar²

2016

IJMTT

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In this article, a new method of analysis of the second order multivariable state-space and singular systems using He's Variational Iteration Method (HVIM) is presented. To illustrate the effectiveness of the He's Variational Iteration Method, an example of the second order multivariable state-space and singular systems has been considered and the solutions were obtained using methods taken from the literature [3] and He's Variational Iteration Method. The obtained discrete solutions are compared with the exact solutions of the second order multivariable state-space and singular systems. Error calculations for the second order multivariable state-space and singular systems have been presented in the Table form to show the efficiency of this He's Variational Iteration Method. This He's Variational Iteration Method can be easily implemented in a digital computer and the solution can be obtained for any length of time. . Keywords -Differential equations, system of differential equations, second order multivariable state-space and singular systems, Leapfrog method, He's Variational Iteration Method.

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“…Cooper and Butcher also show that successive over-relaxation may be applied to improve the rate of convergence for scalar test problem. Gladwell and Thomas [18] recommend this scheme for 2-stage Gauss methods. Each step of the scheme (6) requires s function evaluations and the solution of s sets of n linear equations.…”

Section: Iterative Schemesmentioning

confidence: 99%

Improving Rates of Convergence of Iterative Schemes for Implicit Runge‐Kutta Methods

Vigneswaran

2004

Appl Numer Analy & Comput

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Various iterative schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. In one scheme, when applied to an s-stage Runge-Kutta method, each step of the iteration still requires s function evaluations but consists of r(> s) sub-steps. Improved convergence rate was obtained for the case r = s + 1 only. This scheme is investigated here for the case r = ks, k = 2, 3, · · · , and superlinear convergence is obtained in the limit k → ∞ . Some results are obtained for Gauss methods and numerical results are given.

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Efficiency of Methods for Second-Order Problems

Cited by 15 publications

References 0 publications

Improving Rate of Convergence of an Iterative Scheme With Extra Sub-Steps for Two Stage Gauss Method

Improving Rate of Convergence of an Iterative Scheme With Extra Sub-Steps for Two Stage Gauss Method

Numerical Analysis of second order Multivariable Linear Systems using He’s Variational Iteration Method

Improving Rates of Convergence of Iterative Schemes for Implicit Runge‐Kutta Methods

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