Explicit Runge–Kutta (–Nyström) Methods with Reduced Phase Errors for Computing Oscillating Solutions

Houwen, P. J. van der; Sommeijer, B. P.

doi:10.1137/0724041

Cited by 327 publications

(176 citation statements)

References 19 publications

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“…We mention that the definition of homogeneous dispersion given above is identical to the one used in [8]. Furthermore, our definition of inhomogeneous dispersion is completely analogous to the one used in the literature (see e.g.…”

Section: 2) Dy(l) ()+"° Imentioning

confidence: 56%

“…Finally, we remark that stepsize control can be based on reference formulas of an explicit type that can be constructed along the lines indicated in [8] It is convenient to define the rational functions (in z) Remark. We mention that the definition of homogeneous dispersion given above is identical to the one used in [8].…”

Section: 2) Dy(l) ()+"° Imentioning

confidence: 99%

“…We first derive the (dispersion) relations specifying a qth order dispersive stability function (we remark that for explicit Runge-Kutta methods these relations can be found in [8]). It is shown that there exists a one-parameter family of m-stage, pth order consistent DIRK methods with homogeneous phase errors of order q=2(m-lp/2j).…”

Section: 2) Dy(l) ()+"° Imentioning

confidence: 99%

“…We now use the dispersion relations derived in [8] for stability functions of the form Ill R(z) = 2: f3Jzi.…”

mentioning

confidence: 99%

“…Analogously to a generally adopted approach in the phase-lag analysis of numerical methods for second-order equations with oscillating solutions, we use the equation (cf. [I], [3], (4], [7], [8], [12]- [16]) (…”

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confidence: 99%

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Phase-Lag Analysis of Implicit Runge–Kutta Methods

Houwen¹,

Sommeijer²

1989

SIAM J. Numer. Anal.

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Abstract. We analyse the phase errors introduced by implicit Runge-Kuttu methods when a linear inhomogeneous test equatiotl is integrated. It is shown that the homogeneous phase errors dominate if long interval integrations are performed. Homogeneous dispersion relations for the special class of DIRK methods are derived and a few high-order dispersive DIRK methods are constructed. These methods are applied to systems of linear ditferential equations with oscillating solutions and compared with the "conventional" DIRK methods of NiHsett and Crouzeix.Key words. numerical analysis, ordinary differential equatiotlS, Runge-Kutta methods, periodic solutions AMS(MOS) subject classification. 65L05 l. Introduction. In this paper, special diagonally implicit Runge-Kutta (DIRK) methods will be constructed for integrating systems of OD Es of the formwhen we know in advance that the solution is oscillating. Analogously to a generally adopted approach in the phase-lag analysis of numerical methods for second-order equations with oscillating solutions, we use the equation (cf.as a test equation. Here w represents a natural (or eigen) frequency of the system and w,, represents the frequency of the forced solution component.In § 2, we start by deriving explicit expressions for the phase lag introduced by general, implicit Runge-Kutta (RK) methods. The phase lag is composed of two parts: the homogeneous phase lag corresponding to the eigenmodes in the solution, and the inhomogeneous phase lag corresponding to the forced solution component. We will show that in calculations over long intervals of integration, the homogeneous phase lag tends to increase linearly, whereas the inhomogeneous phase error is constant. For this reason, we concentrate on the reduction of homogeneous phase errors.In § 3, we introduce the concept of a qth order dispersive stability fimction, and we show that such a stability function generates Runge-Kutta methods that have homogeneous phase errors of order q.From § 4 on, we confine our considerations to DIRK methods. We first derive the (dispersion) relations specifying a qth order dispersive stability function (we remark that for explicit Runge-Kutta methods these relations can be found in [8]). It is shown that there exists a one-parameter family of m-stage, pth order consistent DIRK methods with homogeneous phase errors of order q=2(m-lp/2j). In § 5, the dispersion relations are solved for one-, two-, three-, and four-stage DIRK methods and the resulting stability functions are constructed. These functions are dispersive of order q = 2m. The two-stage stability function turns out to be identical with the stability function of the well-known DIRK method of NQlrsett (10].

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Section: 2) Dy(l) ()+"° Imentioning

confidence: 56%

Section: 2) Dy(l) ()+"° Imentioning

confidence: 99%

Section: 2) Dy(l) ()+"° Imentioning

confidence: 99%

“…We now use the dispersion relations derived in [8] for stability functions of the form Ill R(z) = 2: f3Jzi.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Phase-Lag Analysis of Implicit Runge–Kutta Methods

Houwen¹,

Sommeijer²

1989

SIAM J. Numer. Anal.

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New Numerov-type methods for computing eigenvalues, resonances, and phase shifts of the radial Schr�dinger equation

Simos

1997

Int. J. Quant. Chem.

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A new family of P‐stable two‐step Numerov‐type methods with minimal phase lag are developed for the numerical integration of the eigenvalue‐resonance and phase shift problem of the one‐dimensional Schrödinger equation. A new embedding technique to control the phase‐lag error is introduced. Application to various potentials indicates that these new methods are generally more accurate than other previously developed finite‐difference methods. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 467–475, 1997

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New Numerov‐type methods for computing eigenvalues, resonances, and phase shifts of the radial Schrödinger equation

Simos

1997

Int. J. Quant. Chem.

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ABSTRACT:A new family of P-stable two-step Numerov-type methods with minimal phase lag are developed for the numerical integration of the eigenvalue-resonance and phase shift problem of the one-dimensional Schrodinger equation. A new embedding technique to control the phase-lag error is introduced. Application to various potentials indicates that these new methods are generally more accurate than other previously developed finite-difference methods.

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Explicit Runge–Kutta (–Nyström) Methods with Reduced Phase Errors for Computing Oscillating Solutions

Cited by 327 publications

References 19 publications

Phase-Lag Analysis of Implicit Runge–Kutta Methods

Phase-Lag Analysis of Implicit Runge–Kutta Methods

New Numerov-type methods for computing eigenvalues, resonances, and phase shifts of the radial Schr�dinger equation

New Numerov‐type methods for computing eigenvalues, resonances, and phase shifts of the radial Schrödinger equation

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