Forty‐five years of A‐stability

Butcher, John

doi:10.1063/1.2990945

Cited by 5 publications

(8 citation statements)

References 16 publications

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“…Furthermore, an explicit LMM cannot be A stable (see [20]). The weakening of this property leads to the following definition (see [20]). Unfortunately for LMM, the A-stable methods cannot have an order greater than 2, as stated in [20].…”

Section: Definition 4 a Methods Is Stable If The Stability Region Incmentioning

confidence: 99%

“…The region of absolute stability is the set of points hλ for which the method produces a solution that never increases in magnitude, i.e., y n+1 ≤ y n (see [20]). The chosen step size h should lie within the region of absolute stability of the numerical method.…”

Section: Stability Regionmentioning

confidence: 99%

“…Unfortunately for LMM, the A-stable methods cannot have an order greater than 2, as stated in [20]. Furthermore, an explicit LMM cannot be A stable (see [20]). The weakening of this property leads to the following definition (see [20]).…”

Section: Definitionmentioning

confidence: 99%

“…Furthermore, an explicit LMM cannot be A stable (see [20]). The weakening of this property leads to the following definition (see [20]).…”

Section: Definitionmentioning

confidence: 99%

See 3 more Smart Citations

Fixed Coeficient A(α) Stable Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

2019

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The main contribution in this paper is to construct an implicit fixed coefficient Block Backward Differentiation Formulas denoted as A ( α ) -BBDF with equal intervals for solving stiff ordinary differential equations (ODEs). To avoid calculating the differentiation coefficients at each step of the integration, the coefficients of the formulas will be stored, with the intention of optimizing the performance in terms of precision and computational time. The plots of their A ( α ) stability region are provided, and the order of the method is also verified. The necessary conditions for convergence, such as the consistency and zero stability of the method, are also discussed. The numerical results clearly showed the efficiency of the method in terms of accuracy and execution time as compared to other existing methods in the scientific literature.

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Section: Definition 4 a Methods Is Stable If The Stability Region Incmentioning

confidence: 99%

Section: Stability Regionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

“…Furthermore, an explicit LMM cannot be A stable (see [20]). The weakening of this property leads to the following definition (see [20]).…”

Section: Definitionmentioning

confidence: 99%

See 2 more Smart Citations

Fixed Coeficient A(α) Stable Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

2019

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“…If constant |L| increases as presented in [7] the system (10) becomes "stiff" -explicit numerical methods require smaller integration step size for preserving the stability of computation. It is better to use implicit numerical methods for bigger constant |L| .…”

Section: Semi-analytic Computationsmentioning

confidence: 99%

Stiffness in technical initial problems

Kunovský¹,

Šátek²,

Vopěnka³

et al. 2012

AIP Conference Proceedings

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Technical initial problems are defined as initial problems where the right-hand side functions of the system are those occurring in the technical practice, that is functions generated by adding, multiplying and superposing elementary functions. Such systems can be expanded into systems with only rationals operations on the right-hand sides of the equations. In such a case the Taylor series terms can easily be calculated [1]. Test examples are presented in the paper. Stiffness in technical initial problems can be eliminated by the TKSL software and solved by the direct use of the explicit and implicit Taylor series methods.

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Stability of block backward differentiation formulas method

Nasir

İbrahim²,

Suleiman³

et al. 2015

AIP Conference Proceedings

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In this paper, a technique for finding A(α) -stability interval for the Block Backward Differentiation Formulas (BBDFs) are presented. The stability region is obtained for a single equation and stable step size restriction is indicated for the BBDF of order four, (BBDF(4)), and BBDF of order five, (BBDF( 5)). Numerical examples are given to illustrate the stability interval in choosing a suitable step size in the implementation of the method.

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Forty‐five years of A‐stability

Cited by 5 publications

References 16 publications

Fixed Coeficient A(α) Stable Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

Fixed Coeficient A(α) Stable Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

Stiffness in technical initial problems

Stability of block backward differentiation formulas method

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