Analysis of Discretization Methods for Ordinary Differential Equations

Stetter, Hans J.

doi:10.1007/978-3-642-65471-8

Cited by 381 publications

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“…It is well known, see for instance [20], pp. 181, 182, and [15], that unfavourable combinations of strongly varying h. and J. may cause instability for linear multistep methods.…”

Section: Variable Stepsize Linear Multistep Methodsmentioning

confidence: 99%

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Hundsdorfer¹,

Steininger²

1991

BIT

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Abstract.Gehrenbergstrasse 26, Federal Republic Germany To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiffproblems, provided that the stepsize variation is sufficiently regular.

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“…It is well known, see for instance [20], pp. 181, 182, and [15], that unfavourable combinations of strongly varying h. and J. may cause instability for linear multistep methods.…”

Section: Variable Stepsize Linear Multistep Methodsmentioning

confidence: 99%

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Hundsdorfer¹,

Steininger²

1991

BIT

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“…We rewrite (2) into the form which fits into the framework of [5]. We define a family of discrete operators…”

Section: Matrix-vector Representation Of Lmmsmentioning

confidence: 99%

“…The general definition of stability [1,5] which is capable of handling a wide variety of problems, see, e.g., [2,3], is the following. ∃S ∈ R and ∃n 0 ∈ N such that ∀n ≥ n 0 , ∀u n , v n ∈ X n the estimate…”

Section: Stability Of Lmmsmentioning

confidence: 99%

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Note on the stability of strongly stable linear multistep methods

Mincsovics

2017

AIP Conference Proceedings

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“…Above we tacitly assumed that r is constant. However, the process can also be carried out for variable stepsizes [7,8,[14][15][16] on the coarse grid without any additional difficulty. Finally we remark once more that a prerequisite for success of the extrapolation is that the asymptotics hold and thus that both integrations must be stable.…”

Section: T2 = O((ehi:2)-1)mentioning

confidence: 99%

On the odd-even hopscotch scheme for the numerical integration of time-dependent partial differential equations

Boonkkamp¹,

Verwer²

1987

Applied Numerical Mathematics

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This paper is devoted to the odd-even hopscotch scheme for the numerical integration of time-dependent partial differential equations. Attention is focussed on two aspects. Firstly, via the equivalence to the combined leapfrog-Du Fort-Frankel method we derive the explicit expression of the critical time step for von Neumann stability for a class of multi-dimensional convection-diffusion equations. This expression can be derived directly by applying a useful stability theorem due to Hindmarsh, Gresho and Griffiths [9]. The interesting thing on the critical time step is that it is independent of the diffusion parameter and yet smaller than the critical time step for zero diffusion, but only in the multi-dimensional case. This curious phenomenon does not occur for the one-dimensional problem. Secondly, we consider the drawback of the Du Fort-Frankel accuracy deficiency of the hopscotch scheme. To overcome this deficiency we discuss global Richardson extrapolation in time. This simple device can always be used without reducing feasibility. Numerical examples are given to illustrate the outcome of the extrapolation. IntroductionThe subject of this paper is the odd-even hopscotch (OEH) method for the numerical integration in time of time-dependent partial differential equations (PDEs) (Gordon [2], Gourlay [3][4][5] •• , qa) the (constant) velocity, and f.> 0 a diffusion parameter. When combined with simple central differences the OEH method shows an equivalence to the combined leapfrog-Du Fort-Frankel method. Via this equivalence we derive the explicit expression of the critical time step for von Neumann stability for problem (1.1). This expression is easily found by applying a useful theorem due to Hindmarsh, Gresho and Griffiths [9]. This theorem plays an important role in their stability analysis of the forward Euler-central difference scheme. The interesting thing on the critical time step is that it is independent of f. whereas it is smaller than the critical time step for zero diffusion, but only in the multi-dimensional case. We wish to remark that this pathological behaviour of the leapfrog-Du Fort-Frankel method has been observed earlier (see [13] and the references therein). However, to the best of our knowledge, the explicit expression of the critical time step is new.An immediate consequence of this pathological behaviour is that adding artificial diffusion to the OEH central difference scheme may render the process unstable. This observation is in clear contrast to the common practice which teaches us that introducing artificial diffusion has a stabilizing effect. We note that this remark does not contradict the findings of Gourlay and Morris [6] in their investigation of the OEH scheme for nonlinear shock calculations as they restrict their attention to the one-dimensional case.The second aspect of the OEH method considered in this paper is the drawback we refer to as the Du Fort-Frankel (DFF) accuracy deficiency. The consequence of the DFF deficiency is that convergence takes place for a smaller set o...

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Analysis of Discretization Methods for Ordinary Differential Equations

Cited by 381 publications

References 0 publications

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Note on the stability of strongly stable linear multistep methods

On the odd-even hopscotch scheme for the numerical integration of time-dependent partial differential equations

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