Stability regions of one step mixed collocation methods for

This paper provides a theoretical framework for a new type of phase-fitted and amplification-fitted two-step hybrid (FTSH) methods which is introduced by the author in [H. Van de Vyver, A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems, Internat. J. Modern Phys. C 17 (2006) 663-675]. The methods constitute a modification of dissipative two-step hybrid methods in the sense that two free parameters are added to eliminate the phase-lag and the amplification error. The methods are useful only when a good estimate of the frequency of the problem is known in advance. The parameters depend on the product of the estimated frequency and the stepsize. The algebraic order, zero-stability, stability and phase properties are examined. The theory is illustrated with sixth-order explicit FTSH methods. Numerical results carried out on an assortment of test problems show the relevance of the theory.

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“…Similar stability results for several dissipative RK(N) methods with variable coefficients are previously reported (without a proof) in [6,27].…”

Section: Theorem 4 There Exist Values For and For Which The Primary supporting

confidence: 71%

“…A large number of papers on phase-fitted methods have been published including two-step methods [4,23,32] or Runge-Kutta (-Nyström) methods [1,2,21,29]. Another well-known related technique is exponential fitting [6,11,16,19,28,30,31].…”

Section: Introductionmentioning

confidence: 98%

Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)

Vyver

2007

Journal of Computational and Applied Mathematics

119

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“…Stability results of explicit ARKN methods are collected by Franco [32]. For an extensive study of the stability properties of collocation based EFRKN methods we can refer to Carpentieri and Paternoster [29]. The present author [30] has proved the existence of arbitrary high order P-stable exponentially-fitted RKN methods.…”

Section: Introductionmentioning

confidence: 98%

Stability and phase-lag analysis of explicit Runge–Kutta methods with variable coefficients for oscillatory problems

Vyver

2005

Computer Physics Communications

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“…Moreover the system of ODEs (1) may be stiff, and therefore the method has to be implicit and highly stable. Many numerical methods for the direct integration of (1) appeared in the literature: see for example [2], [3], [7], [10], [11] and the bibliography therein contained. We approximate the exact solution y(x) by seeking the continuous method ( ) of the form = ∝ ( )…”

Section: Introductionmentioning

confidence: 99%

“…The positive integer ≥ 2 denotes the step number of the method (2), which is applied directly to provide the solution to (1).…”

Section: Introductionmentioning

confidence: 99%

On the Derivation of High Order Formulae with Interpolants for Solution of Eight-Point Second Order Ordinary Differential Equations

2013

IOSR-JM

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In this Paper, we extend the idea of collocation of linear multistep methods to develop an eight-pointContinuous Block method of order (7,7,7,7,7,7,7,7) T for direct solution of the second order ordinary differential equations. The methods are derived by interpolating the continuous formulation at = + , = and collocating the first and second derivative of the continuous interpolant at + , = 0,1,2, ( ) and = 2, 3, ( ) respectively. This approach yielded the multi discrete schemes that form a self-starting uniform order 7 block methods. The convergence analysis of the methods were discussed and the absolute stability regions shown. Two numerical experiments were used to demonstrate the efficiency of the new methods.

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Stability regions of one step mixed collocation methods for

Cited by 10 publications

References 10 publications

Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)

Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)

Stability and phase-lag analysis of explicit Runge–Kutta methods with variable coefficients for oscillatory problems

On the Derivation of High Order Formulae with Interpolants for Solution of Eight-Point Second Order Ordinary Differential Equations

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