Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems

Frank, Reinhard; Schneid, J.; Ueberhuber, Christoph W.

doi:10.1137/0722031

Cited by 105 publications

(67 citation statements)

References 7 publications

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Order By: Relevance

“…The time derivatives of u are not zero at the boundary; and then the analysis in Example 3.1 shows that the term i- 5 (1/96) A~ uk 2 > behaves only like i-2 · 5 uniformly in h, leading to a decrease in local order of 2.5 units. The other terms of the local error involve higher powers of -r or lower powers of Ah and therefore suffer from reductions which harm less than that of the i-5 A~ uk 2 ' term.…”

Section: Numerical Illustrationmentioning

confidence: 99%

“…When applied to stiff systems of ODEs, not necessarily semi-discrete PDEs, these schemes also suffer from reduction of the order. This is the central issue of the B-convergence theory developed in [5]. In fact, the MOL paper [12] heavily relies on results from the B-convergence theory, whereas [1] is completely independent of it and concentrates on discretizations of ODEs in Banach space.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations

Sanz‐Serna

Verwer²,

Hundsdorfer³

1986

Numer. Math.

104

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Summary. We address the question of convergence of fully discrete RungeKutta approximations. We prove that, under certain conditions, the order in time of the fully discrete scheme equals the conventional order of the Runge-Kutta formula being used. However, these conditions, which are necessary for the result to hold, are not natural. As a result, in many problems the order in time will be strictly smaller than the conventional one, a phenomenon called order reduction. This phenomenon is extensively discussed, both analytically and numerically. As distinct from earlier contributions we here treat explicit Runge-Kutta schemes. Although our results are valid for both parabolic and hyperbolic problems, the examples we present are therefore taken from the hyperbolic field, as it is in this area that explicit discretizations are most appealing.

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Section: Numerical Illustrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations

Sanz‐Serna

Verwer²,

Hundsdorfer³

1986

Numer. Math.

104

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“…Under appropriate stability assumptions one can easily obtain a stiffness-independent convergence result, involving only derivatives of y(t), with an order of convergence equal to the stage order q, see Section 2. Convergence results of this type are well known for Runge-Kutta methods, see for instance Dekker & Verwer (1984), Frank et al (1985b) and Hairer & Wanner (1991). In the latter reference such results were also derived for multistep Runge-Kutta methods.…”

Section: The First Class Contains the Linear Equations Y'(t) = Ly(t) mentioning

confidence: 99%

On the error of general linear methods for stiff dissipative differential equations

Hundsdorfer

1994

IMA J Numer Anal

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Many numerical methods to solve initial value problems of the form y' =f(t, y) can be written as general linear methods. Classical convergence results for such methods are based on the Lipschitz constant and bounds for certain partial derivatives of /. For stiff problems these quantities may be very large, and consequently the classical order of convergence loses its significance. In this paper we consider bounds for the global errors which are based only on bounds for derivatives of y for linear and non-linear dissipative problems with arbitrary stiffness.

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“…It can also be shown that the local order reduction will not occur in case the initial value problem (1.1) is such that all partial derivatives [ai+ iJ(t, v)/8ti8J] with i,j;;:, 0, (i,j) # (0, 1), are bounded by a moderate constant (cf. [9] for a related result with Runge-Kutta methods). Note that the partial derivative with (i,j) = (0, 1), the Jacobian, is always large for stiff systems, since its norm is proportional to the Lipschitz constant.…”

Section: Local Error Boundsmentioning

confidence: 99%

“…Such bounds have been studied quite extensively for RungeKutta methods, for example in [8], [9] and [10]. Most Runge-Kutta methods suffer from an order reduction in the presence of stiffness, i.e., the order of convergence for stiff problems may be considerably lower than for nonstiff problems, even if the solution u(t) is very smooth.…”

mentioning

confidence: 99%

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Hundsdorfer¹,

Steininger²

1991

BIT

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Abstract.Gehrenbergstrasse 26, Federal Republic Germany To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiffproblems, provided that the stepsize variation is sufficiently regular.

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Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems

Cited by 105 publications

References 7 publications

Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations

Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations

On the error of general linear methods for stiff dissipative differential equations

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

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