A scheme for the implementation of implicit Runge-Kutta methods

Cooper, G. J.; Vignesvaran, R.

doi:10.1007/bf02238800

Cited by 27 publications

(21 citation statements)

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“…Cooper and Vignesvaran [9] showed that D(Y ) = 0 if the sequence {Y m } has a limit Y and f is continuous on R n . They observed that the scheme can be implemented efficiently by updating Cooper and Vignesvaran [9] tested the rate of convergence of this scheme when it is applied to the scalar test problem x ′ = qx with rapid convergence required for all z ∈ C − , where C − = {z ∈ C : Re ≤ 0}. For this test problem, the scheme gives (7) gives…”

Section: Efficient Iteration Schemementioning

confidence: 99%

“…Cooper and Vignesvaran [9] imposed the condition that the iteration matrix M has only one non-zero eigenvalue φ,…”

Section: Efficient Iteration Schemementioning

confidence: 99%

“…As an alternative to the modified Newton scheme, some iteration schemes with reduced linear algebra costs have been proposed A scheme of this type proposed in [9] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [9] when it is applied to the scalar test differential equation x ′ = qx and the convergence rate depends on the spectral radius of the iteration matrix M (z), a function of z = hq, where h is the step-length. In this scheme, we require the spectral radius of M (z) to be zero at z = 0 and at z = ∞ in the z-plane in order to improve the rate of convergence of the scheme.…”

mentioning

confidence: 99%

“…In other schemes advantage is taken of the special forms of some implicit methods [2], [4], [5], [12]. In another approach, schemes based directly on iterative procedure have been developed [3], [8], [9], [10], [13], [21]. For a singly implicit method, there is a non-singular matrix S so that S −1 AS = λ(I s − L) −1 , where L is zero except for some ones on the sub-diagonal.…”

mentioning

confidence: 99%

“…These s sub-steps are performed in sequence and it is not possible to compute elements of Y m = y m 1 ⊕ y m 2 ⊕ · · ·⊕ y m s until all sub-steps are completed. Cooper and Vignesvaran [9] considered a scheme where these elements are obtained in sequence and the approximation defect is updated after each sub-step completed. Only one vector transformation is needed for each full step so that this scheme is more efficient.…”

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confidence: 99%

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Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

Kajanthan¹

2014

Int. J. of Pure and Appl. Math.

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Several iteration schemes have been proposed to solve the nonlinear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to the modified Newton scheme, some iteration schemes with reduced linear algebra costs have been proposed A scheme of this type proposed in [9] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [9] when it is applied to the scalar test differential equation x ′ = qx and the convergence rate depends on the spectral radius of the iteration matrix M (z), a function of z = hq, where h is the step-length. In this scheme, we require the spectral radius of M (z) to be zero at z = 0 and at z = ∞ in the z-plane in order to improve the rate of convergence of the scheme. New schemes with parameters are obtained for three-stage and four-stage Gauss methods. Numerical experiments are carried out to confirm the results obtained here.

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Section: Efficient Iteration Schemementioning

confidence: 99%

“…Cooper and Vignesvaran [9] imposed the condition that the iteration matrix M has only one non-zero eigenvalue φ,…”

Section: Efficient Iteration Schemementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

Kajanthan¹

2014

Int. J. of Pure and Appl. Math.

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Improving Rates of Convergence of Iterative Schemes for Implicit Runge‐Kutta Methods

Vigneswaran

2004

Appl Numer Analy & Comput

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Various iterative schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. In one scheme, when applied to an s-stage Runge-Kutta method, each step of the iteration still requires s function evaluations but consists of r(> s) sub-steps. Improved convergence rate was obtained for the case r = s + 1 only. This scheme is investigated here for the case r = ks, k = 2, 3, · · · , and superlinear convergence is obtained in the limit k → ∞ . Some results are obtained for Gauss methods and numerical results are given.

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Development of novel ionization chambers for reference dosimetry in electron flash radiotherapy

Liu,

Holmes,

Khan

et al. 2024

Medical Physics

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BackgroundReference dosimetry in ultra‐high dose rate (UHDR) beamlines is significantly hindered by limitations in conventional ionization chamber design. In particular, conventional chambers suffer from severe charge collection efficiency (CCE) degradation in high dose per pulse (DPP) beams.PurposeThe aim of this study was to optimize the design and performance of parallel plate ion chambers for use in UHDR dosimetry applications, and evaluate their potential as reference class chambers for calibration purposes. Three chamber designs were produced to determine the influence of the ion chamber response on electrode separation, field strength, and collection volume on the ion chamber response under UHDR and ultra‐high dose per pulse (UHDPP) conditions.MethodsThree chambers were designed and produced: the A11‐VAR (0.2–1.0 mm electrode gap, 20 mm diameter collector), the A11‐TPP (0.3 mm electrode gap, 20 mm diameter collector), and the A30 (0.3 mm electrode gap, 5.4 mm diameter collector). The chambers underwent full characterization using an UHDR 9 MeV electron beam with individually varied beam parameters of pulse repetition frequency (PRF, 10–120 Hz), pulse width (PW, 0.5–4 µs), and pulse amplitude (0.01–9 Gy/pulse). The response of the ion chambers was evaluated as a function of the DPP, PRF, PW, dose rate, electric field strength, and electrode gap.ResultsThe chamber response was found to be dependent on DPP and PW, and these dependencies were mitigated with larger electric field strengths and smaller electrode spacing. At a constant electric field strength, we measured a larger CCE as a function of DPP for ion chambers with a smaller electrode gap in the A11‐VAR. For ion chambers with identical electrode gap (A11‐TPP and A30), higher electric field strengths were found to yield better CCE at higher DPP. A PW dependence was observed at low electric field strengths (500 V/mm) for DPP values ranging from 1 to 5 Gy at PWs ranging from 0.5 to 4 µs, but at electric field strengths of 1000 V/mm and higher, these effects become negligible.ConclusionThis study confirmed that the CCE of ion chambers depends strongly on the electrode spacing and the electric field strength, and also on the DPP and the PW of the UHDR beam. A significant finding of this study is that although chamber performance does depend on PW, the effect on the CCE becomes negligible with reduced electrode spacing and increased electric field. A CCE of ≥95% was achieved for DPPs of up to 5 Gy with no observable dependence on PW using the A30 chamber, while still achieving an acceptable performance in conventional dose rate beams, opening up the possibility for this type of chamber to be used as a reference class chamber for calibration purposes of electron FLASH beamlines.

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A scheme for the implementation of implicit Runge-Kutta methods

Cited by 27 publications

References 9 publications

Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

Improving Rates of Convergence of Iterative Schemes for Implicit Runge‐Kutta Methods

Development of novel ionization chambers for reference dosimetry in electron flash radiotherapy

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