A-stable one-step methods with step-size control for stiff systems of ordinary differential equations

Alt, René

doi:10.1016/0771-050x(78)90016-5

Cited by 26 publications

(8 citation statements)

References 13 publications

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“…Implicit methods proved to be more efficient for such IVPs and their development remained an interest in the study of stiff ODEs. Several authors have proposed methods that compute one solution value per step (see, for example [1][2][3][4][5][6][7][8][9]) and those that compute more than one solution value (see for example [9][10][11][12][13][14][15][16][17][18]). …”

Section: Introductionmentioning

confidence: 99%

An improved 2–point block backward differentiation formula for solving stiff initial value problems

Musa¹,

Suleiman²,

Ismail³

et al. 2013

AIP Conference Proceedings

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Articles you may be interested inOn the derivation of second order variable step variable order block backward differentiation formulae for solving stiff ODEs AIP Conf. Abstract.A new block method that generates two values simultaneously is developed for the integration of stiff initial value problems. The method is proven to be A -stable and is a super class of the 2 -point block backward differentiation formula (BBDF). A comparison is made between the method, 1 point backward differentiation formula (BDF) and the 2 point BBDF methods. The numerical results indicate that the new method outperformed the 1 point BDF and the 2 point BBDF methods in terms of accuracy and stability. The total number of steps to complete the integration by the 1 point BDF method is reduced to half. Computation time for the method is also competitive.

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Section: Introductionmentioning

confidence: 99%

An improved 2–point block backward differentiation formula for solving stiff initial value problems

Musa¹,

Suleiman²,

Ismail³

et al. 2013

AIP Conference Proceedings

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show abstract

“…Consider the stiff system taken from [14] y 1 (x) = y 2 (x) − y 1 (x) 2 − (1 + x); y 1 (0) = 1, y 2 (x) = 1 − 20(y 2 (x) 2 − (1 + x) 2 ); y 2 (0) = 1, where x ∈ [0, 1]. The theoretical solution of the system is y 1 (x) = 1 1+x , y 2 (x) = 1 + x.…”

Section: Problemmentioning

confidence: 99%

Solving first-order initial-value problems by using an explicit non-standard A -stable one-step method in variable step-size formulation

Ramos

Singh

Kanwar

et al. 2015

Applied Mathematics and Computation

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“…The Lobatto III A formulas (including the trapezoid rule) are among this class, -stable diagonally implicit formulas of special type were first systematically studied by R. Alt in his 1973 Paris thesis, reported in [1,7]. Recently, J. C.…”

Section: L+k1) L+k2mentioning

confidence: 99%

The modified Newton method in the solution of stiff ordinary differential equations

Alexander¹

1991

Math. Comp.

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Abstract. This paper presents an analysis of the modified Newton method as it is used in codes implementing implicit formulae for integrating stiff ordinary differential equations. We prove that near a smooth solution of the differential system, when the Jacobian is essentially negative dominant and slowly varying, the modified Newton iteration is contractive, converging to the locally unique solution-whose existence is hereby demonstrated-of the implicit equations. This analysis eliminates several common restrictive or unrealistic assumptions, and provides insight for the design of robust codes. V l+k1) l+k2There is an initial transient of duration 0(|/l_1|log|A|), after which the term e ' is not active and the solution is as smooth as cos t. Under suitable conditions, see [29] and infra, solutions of the stiff timevarying linear system

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A-stable one-step methods with step-size control for stiff systems of ordinary differential equations

Cited by 26 publications

References 13 publications

An improved 2–point block backward differentiation formula for solving stiff initial value problems

An improved 2–point block backward differentiation formula for solving stiff initial value problems

Solving first-order initial-value problems by using an explicit non-standard A -stable one-step method in variable step-size formulation

The modified Newton method in the solution of stiff ordinary differential equations

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