A review of recent developments in solving ODEs

Gupta, Gopal; Sacks-Davis, Ron; Tescher, Peter E.

doi:10.1145/4078.4079

Cited by 50 publications

(17 citation statements)

References 117 publications

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“…Lambert, 1973;Shampine and Gordon, 1975;Stoer and Bulirsch, 1983;or Hairer et al, 1987), it is assumed that the reader is already familiar with the basic principles and notation. For further reading the review by Gupta et al (1985) is recommended.…”

Section: Introductionmentioning

confidence: 99%

Numerical integration methods for orbital motion

Montenbruck

1992

Celestial Mech Dyn Astr

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The present report compares Runge-Kutta, multistep and extrapolation methods for the numerical integration of ordinary differential equations and assesses their usefulness for orbit computations of solar system bodies or artificial satellites. The scope of earlier studies is extended by including various methods that have been developed only recently. Several performance tests reveal that modem single-and multistep methods can be similarly efficient over a wide range of eccentricities. Multistep methods are still preferable, however, for ephemeris predictions with a large number of dense output points.

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Section: Introductionmentioning

confidence: 99%

Numerical integration methods for orbital motion

Montenbruck

1992

Celestial Mech Dyn Astr

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“…The use of a special class of multistep formulae, the so-called BDF, was found to be appropriate especially in the case of stiff differential equations (see, for instance, [3] or [13]). The explicit (resp., implicit) BDF of order k are obtained as follows.…”

Section: Adaptive Bdfmentioning

confidence: 99%

“…Our notation refers to that of Hairer We will show P 1 > 0, P 2 > 0 and P 3 < 0 (see (13)). Consequently, our proof consists of three steps.…”

Section: -Stabilitymentioning

confidence: 99%

A-BDF: A Generalization of the Backward Differentiation Formulae

Fredebeul¹

1998

SIAM J. Numer. Anal.

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A one parameter family of generalized BDF, the so-called A-BDF, is proposed and analyzed. Formulae of this new family are A 0 -stable up to order 7, especially A-stable in orders 1 and 2 and A(α)-stable in orders 3 to 7, respectively. The nonexistence result of A 0 -stable formulae of order larger than 7, given by [C. W. Cryer, BIT, 12 (1972), pp. 17-25], [D. M. Creedon and J. J. H. Miller, BIT, 15 (1975), pp. 244-249], is extended to the A-BDF. After implementing these formulae in LSODE, a considerably larger set of stiff problems with oscillatory modes can be solved efficiently and accurately. Introduction.Backward differentiation formulae (BDF) are the most popular tools for the numerical solution of stiff ODEs as they have infinite regions of absolute stability. However, due to the particular shape of the stability regions of BDF of orders 3 through 6, instabilities may occur in problems with highly oscillatory modes such as B5 from STIFF DETEST [8]. In particular, the formula of order 6 is avoided in many BDF-codes (such as LSODE [18]): "For q = 6, the stability region is quite restricted in vertical extent near (but to the left of) the origin. Hence, this order is also excluded, and the available orders for the stiff methods are q = 1, 2, . . . , 5" [2].As a consequence of these restrictions, BDF-codes are inefficient for high accuracy demands.In this paper, we propose and analyze a family of generalized BDF that overcomes this deficiency. Formulae of this new family are stable up to order 7. Moreover, a considerably larger set of stiff problems with oscillatory modes can be solved efficiently and accurately at the expense of minor disadvantages in the solution of "ordinary" stiff problems.We choose the following approach. By subtracting the explicit k-step BDF, multiplied by some parameter t ∈ R, from the corresponding implicit k-step formula, both of order k, we obtain a new implicit formula. Subsequently, these formulae are called adaptive BDF or, to denote the dependence on k, "A-BDF k " (see Definition 2.2). Now we concentrate on the influence of the parameter t on accuracy and stability properties of the scheme. Concerning the question of accuracy, it is sufficient to regard the error constants, at least for practical purposes. Hence, the main part of this paper is devoted to the analysis of the stability properties of the new methods. The central results are the following:• The A-BDF k are consistent (at least) of order k for any k ∈ N and t ∈ R.

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“…Solution of the equations is the main computational cost in the analysis of large ODE systems. Implicit methods are commonly used in solving stiff problems in ODEs because of their stability (Fatunla, 1982;Gupta et al, 1985;Butcher, 2000). Lack of stability causes the normally efficient explicit methods to be unsuitable for stiff problems but recently many authors introduced and developed explicit methods to solve stiff problems (Ahmad et al, 2004;Ahmad and Yaacob, 2005;Hairer et al, 1993;Lambert, 1973;Novati, 2003;Otunta and Ikhile, 1999;Egbako and Adeboye, 2012;Wu and Xia, 2001;2007;Niekerk, 1987;1988;Wu, 1998).…”

Section: Introductionmentioning

confidence: 99%

A Non-Linear Absolutely-Stable Explicit Numerical Integration Algorithm for Stiff Initial-Value Problems

El‐Zahar

2013

American Journal of Applied Sciences

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The time-step in integration process has two restrictions. The first one is the time step restriction due to accuracy requirement τ ac and the second one is the time-step restriction due to stability requirement τ st . The most of explicit methods have small stability regions and consequently small τ st . It obliges us to solve stiff problems with small step size τ st << τ ac . The implicit methods work well with stiff problems but these methods require more work per step than the explicit methods. In this study, a non-linear absolutly stable explicit one step numerical integration algorithm is proposed for solving non linear stiff initial-value problems in ordinary differential equations. The algorithm is based on deriving a non-linear relation between the dependent variable and its derivatives from the well known Taylor expansion. The accuracy of the method depends on some unknown parameter inserted in Taylor expansion and determined from the error analysis. The accuracy and stability properties of the method are investigated and shown to yield at least thirdorder and A-stable. The results obtained in the numerical experiments show the efficiency of the present method in solving stiff initial value problems.

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A review of recent developments in solving ODEs

Cited by 50 publications

References 117 publications

Numerical integration methods for orbital motion

Numerical integration methods for orbital motion

A-BDF: A Generalization of the Backward Differentiation Formulae

A Non-Linear Absolutely-Stable Explicit Numerical Integration Algorithm for Stiff Initial-Value Problems

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