Stability of collocation-based Runge-Kutta-Nyström methods

Houwen, P. J. van den; Sommeijer, B. P.; Công, Nguyễn Hữu

doi:10.1007/bf01933263

Cited by 65 publications

(56 citation statements)

References 11 publications

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“…One can even show that they are a particular form of the SBP-SAT schemes with diagonal norm, see [5]. When applied to second order problems, this leads to the so-called indirect Runge-Kutta-Nyström approach [11,24] with s + 1 implicit stages. We denote these methods GL(2s, s) for s = 1, 2, 3, 4.…”

Section: A Comparison With Other Methodsmentioning

confidence: 99%

Summation-By-Parts in Time: The Second Derivative

Nordström¹,

Lundquist²

2016

SIAM J. Sci. Comput.

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Abstract. We analyze the extension of summation-by-parts operators and weak boundary conditions for solving initial boundary value problems involving second derivatives in time. A wide formulation is obtained by first rewriting the problem on first order form. This formulation leads to optimally sharp fully discrete energy estimates, are unconditionally stable and high order accurate. Furthermore, it provides a natural way to impose mixed boundary conditions of Robin type including time and space derivatives.We apply the new formulation to the wave equation and derive optimal fully discrete energy estimates for general Robin boundary conditions, including non-reflecting ones. The scheme utilizes wide stencil operators in time, whereas the spatial operators can have both wide and compact stencils. Numerical calculations verify the stability and accuracy of the method. We also include a detailed discussion on the added complications when using compact operators in time and give an example showing that an energy estimate cannot be obtained using a standard second order accurate compact stencil.Key words. time integration, second derivative approximation, initial value problem, high order accuracy, wave equation, second order form, initial boundary value problems, boundary conditions, stability, convergence, finite difference, summation-by-parts operators, weak initial conditions AMS subject classifications. 65L20,65M061. Introduction. Hyperbolic partial differential equations on second order form appear in many fields of applications including electromagnetics, acoustics and general relativity, see for example [1,17] and references therein. The most common way of solving these equations has traditionally been to rewrite them on first order form and apply well-established methods for solving first order hyperbolic problems. For the time integration part, a popular choice due to its time reversibility is the leapfrog method, especially for long time simulations of wave propagation problems. This traditional approach does however have some disadvantages. Most notably it increases the number of unknowns and requires a higher resolution in both time and space [16]. Many attempts have been made to instead discretize the second order derivatives directly with finite difference approximations. Various types of compact difference schemes have been proposed, e.g. including the classical methods of Störmer and Numerov [11,21,13], as well as one-step methods of Runge-Kutta type [11].Of particular interest to us is the development of schemes based on high order accurate operators in space obeying a summation-by-parts (SBP) rule together with the weak simultaneous-approximation-term (SAT) penalty technique for boundary conditions, [7,15]. The SBP-SAT technique in space in combination with well posed boundary conditions leads to energy stable semi-discrete schemes. By augmenting the SBP-SAT technique in space with SBP-SAT in time, we arrive at a procedure that leads in an almost automatic way to fully discrete schemes with uncon...

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Section: A Comparison With Other Methodsmentioning

confidence: 99%

Summation-By-Parts in Time: The Second Derivative

Nordström¹,

Lundquist²

2016

SIAM J. Sci. Comput.

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“…It should be noted that all classical collocation RKN methods are implicit (cf. [18]). It is well known that implicit RK(N) methods should only be used for solving stiff problems.…”

Section: Explicit Pseudo Two-step Rkn (Eptrkn) Methodsmentioning

confidence: 99%

“…It has been of interest to develop methods to solve equation (1) without rewriting it as a system of first-order ODEs. Numerical methods for solving equation (1) directly have been developed extensively in the literature (see, e.g., [2], [3], [5], [6], and [18]). Among direct methods for solving (1), Runge-Kutta-Nyström (RKN) methods are preferable.…”

Section: Introductionmentioning

confidence: 99%

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

Hoang

2015

Applied Numerical Mathematics

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A general class of functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström (FEPTRKN) methods for solving second-order initial value problems has been studied. These methods can be considered generalized explicit pseudo two-step Runge-Kutta-Nyström (EPTRKN) methods. We proved that an s-stage FEPTRKN method has step order p = s and stage order r = s for any set of distinct collocation parameters (ci). Supperconvergence for the accuracy orders of these methods can be obtained if the collocation parameters (ci) s i=1 satisfy some orthogonality conditions. We proved that an s-stage FEPTRKN method can attain accuracy order p = s + 3. Numerical experiments have shown that the new FEPTRKN methods work better than do EPTRKN methods on problems whose solutions can be well approximated by the functions in bases on which these FEPTRKN methods are developed.

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“…Hence, to converge as fast to the corrector solution as possible, a simple analysis based on the model equation y" = Ay indicates that we should minimize the spectral radius of the iteration matrix h 2 AA. In [19], Nguyen huu Cong studied this subject in more detail and found that RKN correctors based on so-called "direct collocation" (see also [16]) possess a smaller convergence factor (i.e., p(A)) than the "indirect collocation"-based correctors that we used in the present paper. Although these "direct RKN correctors" are usually not unconditionally stable, their stability boundaries are in many cases sufficiently large for nonstiff problems.…”

Section: The Rate Of Convergencementioning

confidence: 99%

Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

Sommeijer¹

1993

Applied Numerical Mathematics

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Sommeijer, B.P., Explicit, high-order Runge-Kutta-Nystrom methods for parallel computers, Applied Numerical Mathematics 13 (1993) 221-240.The paper describes the construction of explicit Runge-Kutta-Nystri:im (RKN) methods of arbitrarily high order. The order is borrowed from an underlying implicit RKN method. For the approximate solution of this method, an iteration scheme is defined. Prescribing a fixed number of iterations, the resulting scheme is an explicit RKN method. The iteration scheme is defined in such a way that many of the right-hand side evaluations can be done concurrently. As a result, explicit RKN schemes of order p are obtained which require, on a parallel computer, approximately p /2 right-hand side evaluations per step. Both in fixed-and variable-step mode, the schemes are compared with existing (sequential) high-order RKN methods from the literature and are shown to demonstrate superior behaviour.

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Stability of collocation-based Runge-Kutta-Nyström methods

Cited by 65 publications

References 11 publications

Summation-By-Parts in Time: The Second Derivative

Summation-By-Parts in Time: The Second Derivative

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

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