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1.
In this paper a generator of hybrid methods with minimal phase-lag is developed for the numerical solution of the Schrödinger equation and related problems. The generator's methods are dissipative and are of eighth algebraic order. In order to have minimal phase-lag with the new methods, their coefficients are determined automatically. Numerical results obtained by their application to some well known problems with periodic or oscillating solutions and to the coupled differential equations of the Schrödinger type indicate the efficiency of these new methods.  相似文献   

2.
Runge-Kutta-Nyström type methods and special predictor-corrector methods are constructed for the accurate solution of second-order differential equations of which the solution is dominated by the forced oscillation originating from an external, periodic forcing term. For a family of second-order explicit and linearly implicit Runge-Kutta-Nyström methods it is shown that the forced oscillation is represented with zero phase lag. For a family of predictor-corrector methods of fourth-order, it is shown that both the phase lag order and the dissipation of the forced oscillation can be made arbitrarily high. Numerical examples illustrate the effectiveness of our reduced phase lag methods.  相似文献   

3.
An explicit symmetric multistep method is presented in this paper. The new method is exponentially fitted and trigonometrically-fitted and is of algebraic order eight. The effectiveness of the exponential fitting is proved by the application of the new method and the classical one (with constant coefficients) to well-known periodic problems.  相似文献   

4.
《国际计算机数学杂志》2012,89(1-2):135-140
A two-step method with phase-lag of order infinity is developed for the numerical integration of second order periodic initial-value problem. The method has algebraic order six. Extensive numerical testing indicates that the new method is generally more accurate than other two-step methods.  相似文献   

5.
6.
In this paper we present a new optimized symmetric eight-step predictor-corrector method with phase-lag of order infinity (phase-fitted). The method is based on the symmetric multistep method of Quinlan–Tremaine, with eight steps and eighth algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation during the resonance problem with the use of the Woods–Saxon potential. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.  相似文献   

7.
New Runge-Kutta-Nyström methods especially designed for the numerical integration of perturbed oscillators are presented in this paper. They are capable of exactly integrating the harmonic or unperturbed oscillator. We construct an embedded 4(3) RKN pair that is based on the FSAL property. The new method is much more efficient than previously derived RKN methods for some subclasses of problems.  相似文献   

8.
This paper is concerned with the long term behaviour of the error generated by one step methods in the numerical integration of periodic flows. Assuming numerical methods where the global error possesses an asymptotic expansion and a periodic flow with the period depending smoothly on the starting point, some conditions that ensure an asymptotically linear growth of the error with the number of periods are given. A study of the error growth of first integrals is also carried out. The error behaviour of Runge–Kutta methods implemented with fixed or variable step size with a smooth step size function, with a projection technique on the invariants of the problem is considered.  相似文献   

9.
In order to improve the efficiency and accuracy of the previous Obrechkoff method, in this paper we put forward a new kind of P-stable three-step Obrechkoff method of O(h10) for periodic initial-value problems. By using a new structure and an embedded high accurate first-order derivative formula, we can avoid time-consuming iterative calculation to obtain the high-order derivatives. By taking advantage of new trigonometrically-fitting scheme we can make both the main structure and the first-order derivative formula to be P-stable. We apply our new method to three periodic problems and compare it with the previous three Obrechkoff methods. Numerical results demonstrate that our new method is superior over the previous ones in accuracy, efficiency and stability.  相似文献   

10.
The paper provides an overview on the use of high-resolution methods based on the CABARET scheme. The results are provided for several test problems including gas dynamics, computational aeroacoustics, and geophysical fluid dynamics for a classical double-gyre quasi-geostrophic model of ocean dynamics.  相似文献   

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12.
A new Runge-Kutta-Nyström method, with phase-lag of order infinity, for the integration of second-order periodic initial-value problems is developed in this paper. The new method is based on the Dormand, El-Mikkawy and Prince Runge-Kutta-Nyström method of algebraic order four with four (three effective) stages. Numerical illustrations indicate that the new method is much more efficient than other methods derived, based on the idea of minimal phase lag or of phase lag of order infinity.  相似文献   

13.
We have proposed exposure method using Deep UV (DUV) exposure light to realize high resolution and high productivity in FPD lithography. Here, we show development concepts for our new G6 exposure tools with DUV light sources and exposure test results with a test exposure tool.  相似文献   

14.
For the numerical solution of initial value problems a general procedure to determine global integration methods is derived and studied. They are collocation methods which can be easily implemented and provide a high order accuracy. They further provide globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. Numerical experiments provide favorable comparisons with other existing methods.  相似文献   

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Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order  相似文献   

17.
In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.  相似文献   

18.
In a recent paper, an error estimate of a one-step numerical method, originated from the Lanczos tau method, for initial value problems for first order linear ordinary differential equations with polynomial coefficients, was obtained, based on the error of the Lanczos econo-mization process. Numerical results then revealed that the estimate gives, correctly, the order of the tau approximant being sought. In the present paper we further establish that the error estimate is optimum with respect to the integration of the error equation. Numerical examples are included for completeness.  相似文献   

19.
《国际计算机数学杂志》2012,89(8):1027-1038
A second-order, unconditionally-stable, finite-difference scheme is developed for the numerical solution of the SI model of fox-rabies dynamics. The local stability of the scheme, by direct inspection of the eigenvalues dependent on the time step size and on two parameters, is shown to be unconditionally stable.  相似文献   

20.
《国际计算机数学杂志》2012,89(15):3324-3334
In this paper, we present a class of one-step explicit zero-dissipative nonlinear methods for the numerical integration of perturbed oscillators, which have second algebraic order and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.  相似文献   

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