Interpolants for Runge-Kutta formulas

Enright, W. H.; Jackson, Kenneth R.; Nørsett, Syvert P.; Thomsen, Per Grove

doi:10.1145/7921.7923

Cited by 196 publications

(106 citation statements)

References 9 publications

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“…For the 6th-order case, a particular strategy is used to obtain a family of schemes. The particular families presented in this paper for the 4th-, 5th-, and 6th-order cases are interesting because they can be viewed as instances of continuous schemes derived from the more traditional interpolation-based approach for the construction of continuous extensions to Runge-Kutta schemes (Enright et al [17]). In each of these cases, the stages which do not satisfy C(p -1), for the /rth-order family, have zero weight polynomials.…”

Section: Discussionmentioning

confidence: 99%

“…A number of authors have demonstrated the possibility of generating inexpensive interpolants for explicit Runge-Kutta formulas which are not of the collocation type, in the context of the numerical solution of initial value problems (see, for example, Enright et al [17], Gladwell et al [20], and references therein for some of the earlier work). These interpolants are obtained for Runge-Kutta formula pairs, by constructing extra stages within the current step, thus preserving the one-step nature of the formula.…”

Section: Introductionmentioning

confidence: 99%

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Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes

Muir¹,

Owren²

1993

Math. Comp.

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Abstract. The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODEs more than fifteen years ago. During the last decade, a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ODE problems, where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes. Recent work in this area has seen the development of some software packages for boundary value ODEs based on these schemes. Unfortunately, these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval, while the collocation schemes provide a natural continuous solution approximation. The availability of a continuous solution is important not only to the user of the software but also within the code itself, for example, in estimation of errors, defect control, mesh selection, and the provision of initial solution estimates for new meshes. An approach for the construction of a continuous solution approximation based on the MIRK schemes is suggested by recent work in the area of continuous extensions for explicit Runge-Kutta schemes for initial value ODEs. In this paper, we describe our work in the investigation of continuous versions of the MIRK schemes: (i) we give some lower bounds relating the stage order to the minimal number of stages for general continuous IRK schemes, (ii) we establish lower bounds on the number of stages needed to derive continuous MIRK schemes of orders 1 through 6, and (iii) we provide characterizations of these schemes having a minimal number of stages for each of these orders.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes

Muir¹,

Owren²

1993

Math. Comp.

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“…There are different approaches that can be used to derive SDC CRK formulas. We will use a "bootstrapping" approach similar to that introduced in [6] to derive higher order local interpolants. Other approaches, such as the direct solution of the continuous order conditions, are also possible and could lead to a wider class of suitable SDC CRK formulas.…”

Section: Deriving An Sdc Crkmentioning

confidence: 99%

“…From (6) and (7) we see that as h i → 0 the defect will behave like a linear combination of theq j (τ) over each [x i−1 , x i ]. In the special case that L = 1 the shape of the defect will be the same (as h i → 0) for all problems and all steps.…”

Section: Introductionmentioning

confidence: 99%

The reliability/cost trade-off for a class of ODE solvers

Enright

2009

Numer Algor

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In the numerical solution of ODEs, it is now possible to develop efficient techniques that will deliver approximate solutions that are piecewise polynomials. The resulting methods can be designed so that the piecewise polynomial will satisfy a perturbed ODE with an associated defect (or residual) that is directly controlled in a consistent fashion. We will investigate the reliability/cost trade off that one faces when implementing and using such methods, when the methods are based on an underlying discrete Runge-Kutta formula.In particular we will identify a new class of continuous Runge-Kutta methods with a very reliable defect estimator and a validity check that reflects the credibility of the estimate. We will introduce different measures of the "reliability" of an approximate solution that are based on the accuracy of the approximate solution; the maximum magnitude of the defect of the approximate solution; and how well the method is able to estimate the maximum magnitude of the defect of the approximate solution. We will also consider how methods can be implemented to detect and cope with special difficulties such as the effect of round-off error (on a single step) or the ability of a method to estimate the magnitude of the defect when the stepsize is large (as might happen when using a high-order method at relaxed accuracy requests).Numerical results on a wide selection of problems will be summarized for methods of orders five, six and eight. It will be shown that a modest increase in the cost per step can lead to a significant improvement in the quality of the approximate solutions and the reliability of the method. For example, the numerical results demonstrate that, if one is willing to increase the cost per step by 50%, then a method can deliver approximate solutions where the reported estimated maximum defect is within 1% of its true value on 95% of the steps.

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“…For many applications it is now recognized that an accurate discrete approximation is not enough and most numerical methods now provide an accurate approximation to the solution of (1) that can be evaluated at any value of x ∈ [a, b]. For a discussion of how this is done and how such methods are used see [10], [3] and [7]. In particular Figures 1 and 2 show the advantage such a method has when it is used to display (or visualize) the solution of an IVP.…”

Section: Introductionmentioning

confidence: 99%

Reducing the Uncertainty When Approximating the Solution of ODEs

Enright

2012

IFIP Advances in Information and Communication Technology

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Abstract. One can reduce the uncertainty in the quality of an approximate solution of an ordinary differential equation (ODE) by implementing methods which have a more rigorous error control strategy and which deliver an approximate solution that is much more likely to satisfy the expectations of the user. We have developed such a class of ODE methods as well as a collection of software tools that will deliver a piecewise polynomial as the approximate solution and facilitate the investigation of various aspects of the problem that are often of as much interest as the approximate solution itself. We will introduce measures that can be used to quantify the reliability of an approximate solution and discuss how one can implement methods that, at some extra cost, can produce very reliable approximate solutions and therefore significantly reduce the uncertainty in the computed results.

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Interpolants for Runge-Kutta formulas

Cited by 196 publications

References 9 publications

Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes

Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes

The reliability/cost trade-off for a class of ODE solvers

Reducing the Uncertainty When Approximating the Solution of ODEs

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