Order Stars and Linear Stability Theory

Sofroniou, Mark

doi:10.1006/jsco.1996.0004

Cited by 13 publications

(9 citation statements)

References 17 publications

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Order By: Relevance

“…As it can be observed, the numerical estimation of accuracy order agrees with the theoretical prediction. Note that these results confirm that the definition of the load term, given in Equation (45), is actually consistent, since the desired accuracy is retained up to the last stage.…”

Section: Test 1: a Harmonically Excited Oscillator With Viscous Dampingsupporting

confidence: 82%

“…An elegant analysis can be performed by a powerful modern tool called order star, which conveys important information such as accuracy and stability properties in a unified framework [43][44][45]. The crucial idea is to extend standard stability, Equation (11), by comparing the approximate amplification factor A to the exact amplification factor (instead to 1)…”

Section: Algorithmic Propertiesmentioning

confidence: 99%

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Hierarchical higher-order dissipative methods for transient analysis

Govoni

Mancuso

Ubertini

2006

Int. J. Numer. Meth. Engng

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SUMMARYThis work focuses on devising an efficient hierarchy of higher-order methods for linear transient analysis, equipped with an effective dissipative action on the spurious high modes of the response. The proposed strategy stems from the Nørsett idea and is based on a multi-stage algorithm, designed to hierarchically improve accuracy while retaining the desired dissipative behaviour. Computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved (as standard time integration methods) and, in addition, all the stages should share the same coefficient matrix. This target is achieved by rationally formulating the methods based on the discontinuous collocation approach. The resultant procedure is shown to be well suited for adaptive solution strategies. In particular, it embeds two natural tools to effectively control the error propagation. One estimates the local error through the next-stage solution, which is one-order more accurate, the other through the solution discontinuity at the beginning of the current time step, which is permitted by the present formulation. The performance of the procedure and the quality of the two error estimators are experimentally verified on different classes of problems. Some typical numerical tests in transient heat conduction and elasto-dynamics are presented.

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Section: Test 1: a Harmonically Excited Oscillator With Viscous Dampingsupporting

confidence: 82%

Section: Algorithmic Propertiesmentioning

confidence: 99%

Hierarchical higher-order dissipative methods for transient analysis

Govoni

Mancuso

Ubertini

2006

Int. J. Numer. Meth. Engng

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“…[17]). Figures 5 and 6 show the stability regions and order stars, respectively, of the QM θ -methods calculated using the following expression for the stability root…”

Section: Qm Non-linear θ-Methodsmentioning

confidence: 93%

“…, yielding two vertical branch cuts, not shown in the stability regions and order stars in Figures 1 and 2, respectively, drawn by using Sofroniou's OrderStar Mathematica package [17], the present version of which does not deal with branch points and cuts. The function used as argument for OrderStar was the so-called stability root…”

Section: Sivakumar-savithri Ham Non-linear θ-Methodsmentioning

confidence: 99%

Stability by order stars for non-linear theta-methods based on means

Villatoro

2010

International Journal of Computer Mathematics

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The linear stability analysis of non-linear one-step methods based on means is studied by means of the concept of stability regions and order stars. Concretely, non-linear θ -methods based on harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are considered. Their stability diagrams and order stars show their A-stability for θ ≥ 1/2, and L-stability in some cases. Order stars in the Riemann surface are a requirement for non-linear one-step methods. The advantages and disadvantages of this technique are presented.

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“…They were obtained, using the MAPLE computer algebra system, by brute force identification of the coefficients of the Taylor series of the exact solution y(x 0 +h) and the ones of the result of the Runge-Kutta formula, denoted y 1 (h) in [21, II.1, Stability Analysis From a theoretical point of view, the stability of Butcher tableaux can be determined by computing the stability function R(z) of each tableau and establishing that its stability domain -which is the subset of the complex plane such that |R(z)| < 1 -is non empty. Some existing computer algebra software are dedicated to this study [38] but we could not take advantage of them by lack of access to Mathematica. Instead, we directly computed R(z) using [22, IV, (2.8)].…”

Section: The Methodsmentioning

confidence: 99%