Generalized Multistep Predictor-Corrector Methods

Gragg, William B.; Stetter, Hans J.

doi:10.1145/321217.321223

Cited by 161 publications

(89 citation statements)

References 4 publications

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“…Then we have the following result due to Gragg and Stetter [1964]. T 5.1 ( [Gragg and Stetter, 1964]).…”

Section: Multirate Base Methodsmentioning

confidence: 81%

“…Then we have the following result due to Gragg and Stetter [1964]. T 5.1 ( [Gragg and Stetter, 1964]). Suppose that a given method with sufficiently smooth increment function Φ satisfies the consistency condition Φ x, y, 0 = f (x, y) and possesses an expansion (5.2) for the local error.…”

Section: Multirate Base Methodsmentioning

confidence: 81%

“…Following [Gragg and Stetter, 1964;Hairer et al, 1993a] we are looking for a global error function e p (x) of the form (see [Hairer et al, 1993a, Chp. II, Thm.…”

Section: Multirate Base Methodsmentioning

confidence: 99%

See 2 more Smart Citations

On Extrapolated Multirate Methods

Constantinescu

Sandu

2010

Progress in Industrial Mathematics at ECMI 2008

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Abstract. In this manuscript we construct extrapolated multirate discretization methods that allow to efficiently solve problems that have components with different dynamics. This approach is suited for the time integration of multiscale ordinary and partial differential equations and provides highly accurate discretizations. We analyze the linear stability properties of the multirate explicit and linearly implicit extrapolated methods. Numerical results with multiscale ODEs illustrate the theoretical findings.

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“…Then we have the following result due to Gragg and Stetter [1964]. T 5.1 ( [Gragg and Stetter, 1964]).…”

Section: Multirate Base Methodsmentioning

confidence: 81%

Section: Multirate Base Methodsmentioning

confidence: 81%

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On Extrapolated Multirate Methods

Constantinescu

Sandu

2010

Progress in Industrial Mathematics at ECMI 2008

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“…J. C. Butcher [4], C. W. Gear [19], Gragg and Stetter [20] and others) for L < K + 1. In view of the regularity of the partitions and hypothesis (3.5) we can prove the following useful lemma.…”

Section: E) From This Fact and Hypotheses (Ii) And (Iii) Of Theoremmentioning

confidence: 94%

Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations

Delfour¹,

Dubeau²

1986

Math. Comp.

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Abstract. This paper studies the approximation of the solution of nonlinear ordinary differential equations by (discontinuous) piecewise polynomials of degree K and traces at the nodes of discretization. A mesh-dependent variational framework underlying this discontinuous approximation is derived. Several families of one-step, hybrid and multistep schemes are obtained. It is shown that the convergence rate in the ¿2-norm is K + 1. The nodal-convergence rate can go up to 2 K + 2, depending on the particular scheme under consideration. The mesh-dependent variational framework introduced here is of special interest in the approximation of the solution of optimal control problems governed by differential equations.1. Introduction. The object of this paper is the study of (discontinuous) piecewise polynomial approximations to the solution of systems of nonlinear ordinary differential equations defined on a fixed interval [0, T], T > 0. The type of approximation we shall use can be briefly described in the following way. The interval [0, T] is first partitioned into TV intervals by specifying a sequence {t"}^0, 0 = t0 < tx < ■ ■ ■ < tN = T, of real numbers. On each interval /" = [t"_x, t"], n = 1,..., TV, we construct a polynomial u" in PK(In), the space of polynomials of degree at most K > 0 defined on the interval /". At each node t", we specify a trace (or a point) U", n = 0,..., TV. So the approximation problem consists in finding the TV polynomials {un}"=x and the TV + 1 traces (or points) {[7"}^=0. We shall see that this kind of approximation leads to a global //-convergence rate of K + 1 and a nodal-convergence rate (for the traces {t/"}'s) of 2K + 2.In this paper we adopt a more general formulation, of which the above-described approximation is a special case. On each interval /" we allow L, 0 < L < K + 1, additional conditions on each polynomial un in PK(In). For instance, when L = 2 and «"('"_i)= U"_x and u"(t") = U", we obtain the continuous piecewise-polynomial approximation of B. L. Hulme [22], [23]. For that method, the global L2-convergence rate is K + 1 and the convergence rate at the nodes is 2K. This framework encompasses the a-method of Delfour, Hager and Trochu [14] for L = 1, and a"un(t") + (1 -an)u"+x(t") = X", n = 0,...,TV, where {a"}N=0 is an a priori

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“…While it is possible to modify these theories to include various methods which fall between these extreme types ( [5], [6], [7], [8], [9]), it is of interest to analyse the behaviour of error propagation for a class of method of sufficient generality to include all these special cases in a natural and straightforward way. In fact, we will consider the type of general method formulated by the author [10].…”

Section: Introductionmentioning

confidence: 99%