An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs

Jackson, Kenneth R.; Sacks-Davis, Ron

doi:10.1145/355900.355903

“…More details on how this expression is obtained are given in [3], based on the work of Jackson and Sack-Davis [13]. Using (7), computing y n+1 amounts to solving the nonlinear system G t n+1 , y n+1 , y…”

Section: Bdf Discretizationmentioning

confidence: 99%

Strategies for solving index one DAE with non-negative constraints: Application to liquid–liquid extraction

Métivier

¹

,

Montarnal

²

2012

Journal of Computational Physics

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“…The normalization «o« = «o will be assumed. Any other normalization may not always be applicable; for example, the more natural normalization Y.ß = 1 sometimes fails for the / variant of the fixed leading coefficient formulas of Jackson and Sacks-Davis [11]. Requiring that the formula be exact for polynomials of degree at most q imposes another q + 1 linear conditions on the coefficients.…”

Section: Statesmentioning

confidence: 99%

Construction of variable-stepsize multistep formulas

Skeel¹

1986

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Abstract. A systematic way of extending a general fixed-stepsize multistep formula to a minimum storage variable-stepsize formula has been discovered that encompasses fixed-coefficient (interpolatory), variable-coefficient (variable step), and fixed leading coefficient as special cases. In particular, it is shown that the " interpolatory" stepsize changing technique of Nordsieck leads to a truly variable-stepsize multistep formula (which has implications for local error estimation and formula changing), and it is shown that the " variable-step" stepsize changing technique applicable to the Adams and backward-differentiation formulas has a reasonable generalization to the general multistep formula. In fact, it is shown how to construct a variable-order family of variable-coefficient formulas. Finally, it is observed that the first Dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration.1. Introduction. Multistep methods have been the most successful numerical methods for solving initial-value problems in ordinary differential equations. The selection of a particular formula is often based on a theoretical analysis of fixedstepsize formulas, and yet implementation normally requires the use of a variablestepsize formula. The question of how to extend a formula to variable stepsize is the primary topic of this paper. Existing techniques for varying stepsize are studied, revealing interesting relationships and useful generalizations. At the same time, the results in Skeel [19] on the equivalence between multivalue methods and multistep methods are extended to variable stepsize. There are techniques for varying the stepsize other than the use of variable-stepsize formulas, and these are included in

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“…The natural extensions referred to here are implemented in such codes as EPISODE (Byrne and Hindmarsh [1]) and ODE/DE/STEP, INTRP (Shampine and Gordon [17]), and have been named "variable-coefficient" formulas by Jackson and SacksDavis [11]. In Section 5 of this paper we present a systematic way of extending fixed-stepsize formulas to variable-stepsize formulas of which the natural variablecoefficient Adams and backward-differentiation formulas are particular cases.…”

Section: Statesmentioning

confidence: 99%

“…This technique for varying the stepsize is applicable to general fixed-stepsize formulas. The precise nature of this type of variable-stepsize method has been somewhat of a mystery, but if there is a prevailing view, then it is most clearly expressed by Jackson and Sacks-Davis [11]:…”

Section: Statesmentioning

confidence: 99%

Construction of Variable-Stepsize Multistep Formulas

Skeel¹

1986

Mathematics of Computation

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Abstract. A systematic way of extending a general fixed-stepsize multistep formula to a minimum storage variable-stepsize formula has been discovered that encompasses fixed-coefficient (interpolatory), variable-coefficient (variable step), and fixed leading coefficient as special cases. In particular, it is shown that the " interpolatory" stepsize changing technique of Nordsieck leads to a truly variable-stepsize multistep formula (which has implications for local error estimation and formula changing), and it is shown that the " variable-step" stepsize changing technique applicable to the Adams and backward-differentiation formulas has a reasonable generalization to the general multistep formula. In fact, it is shown how to construct a variable-order family of variable-coefficient formulas. Finally, it is observed that the first Dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration.1. Introduction. Multistep methods have been the most successful numerical methods for solving initial-value problems in ordinary differential equations. The selection of a particular formula is often based on a theoretical analysis of fixedstepsize formulas, and yet implementation normally requires the use of a variablestepsize formula. The question of how to extend a formula to variable stepsize is the primary topic of this paper. Existing techniques for varying stepsize are studied, revealing interesting relationships and useful generalizations. At the same time, the results in Skeel [19] on the equivalence between multivalue methods and multistep methods are extended to variable stepsize. There are techniques for varying the stepsize other than the use of variable-stepsize formulas, and these are included in

show abstract

An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs

Cited by 84 publications

References 7 publications

Strategies for solving index one DAE with non-negative constraints: Application to liquid–liquid extraction

Strategies for solving index one DAE with non-negative constraints: Application to liquid–liquid extraction

Construction of variable-stepsize multistep formulas

Construction of Variable-Stepsize Multistep Formulas

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