An Iteration Scheme for Implicit Runge—Kutta Methods

Cooper, G. J.; Butcher, John

doi:10.1093/imanum/3.2.127

Cited by 55 publications

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“…The -stage implicit RK time integration scheme has 2 order of convergence and is A-stable. The interested reader is referred to [44][45][46], for more details about -stage implicit RK method and how to choose the parameters and ε at = 2.…”

Section: Solutions Of the Sodes [(14)-(17) And (35)-(37)]mentioning

confidence: 99%

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

2014

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Abstract:In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach. PACS (2008): 35K10, 65M70, 33C05Keywords: non-local boundary conditions • integral conservation condition • collocation method • shifted Jacobi-GaussLobatto quadrature • system of differential equations © Versita sp. z o.o.

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Section: Solutions Of the Sodes [(14)-(17) And (35)-(37)]mentioning

confidence: 99%

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

2014

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“…He suggested how higher order methods could be used in combination with diagonally implicit methods through an iterated defect correction process. Cooper and Butcher [8] considered a more general scheme which explicitly uses the Jacobian of differential system. Cooper and Vignesvaran [9] developed an alternative scheme which is computationally more efficient.…”

Section: Introductionmentioning

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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“…In other schemes advantage is taken of the special forms of some implicit methods [2], [4], [5], [12]. In another approach, schemes based directly on iterative procedure have been developed [3], [8], [9], [10], [13], [21]. For a singly implicit method, there is a non-singular matrix S so that S −1 AS = λ(I s − L) −1 , where L is zero except for some ones on the sub-diagonal.…”

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confidence: 99%

“…Cooper and Butcher [8] proposed an iterative scheme, sacrificing superlinear convergence for reduced linear algebra cost, which may be regarded as a generalization of the scheme (5) for singly implicit methods. They considered the scheme…”

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Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

Kajanthan¹

2014

Int. J. of Pure and Appl. Math.

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Several iteration schemes have been proposed to solve the nonlinear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to the modified Newton scheme, some iteration schemes with reduced linear algebra costs have been proposed A scheme of this type proposed in [9] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [9] when it is applied to the scalar test differential equation x ′ = qx and the convergence rate depends on the spectral radius of the iteration matrix M (z), a function of z = hq, where h is the step-length. In this scheme, we require the spectral radius of M (z) to be zero at z = 0 and at z = ∞ in the z-plane in order to improve the rate of convergence of the scheme. New schemes with parameters are obtained for three-stage and four-stage Gauss methods. Numerical experiments are carried out to confirm the results obtained here.

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An Iteration Scheme for Implicit Runge—Kutta Methods

Cited by 55 publications

References 0 publications

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

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

Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

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