On pade′ approximations to the exponential function and application to the point kinetics equations

Aboanber, Ahmed E.; Nahla, Abdallah A.

doi:10.1016/j.pnucene.2004.07.003

Cited by 33 publications

(18 citation statements)

References 22 publications

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“…Numerical experiments show the mean-square convergence orders of the proposed schemes. For the deterministic situations, it is known that the numerical schemes based on the Padé approximation are A-stable ( [10]) under appropriate conditions. However, stochastic stability of our methods still need further investigation.…”

Section: Resultsmentioning

confidence: 99%

Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

Sun

Wang

2017

Journal of Computational and Applied Mathematics

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In this article, we introduce a kind of numerical schemes, based on Padé approximation, for two stochastic Hamiltonian systems which are treated separately. For the linear stochastic Hamiltonian systems, it is shown that the applied Padé approximations P (k,k) give numerical solutions that inherit the symplecticity and the proposed numerical schemes based on P (r,s) are of meansquare order r+s 2 under appropriate conditions. In case of the special stochastic Hamiltonian systems with additive noises, the numerical method using two kinds of Padé approximation P (r,ŝ) and P (ř,š) has mean-square orderř +š + 1 whenr +ŝ =ř +š + 2. Moreover, the numerical solution is symplectic if r =ŝ.

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Section: Resultsmentioning

confidence: 99%

Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

Sun

Wang

2017

Journal of Computational and Applied Mathematics

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“…The Hermite polynomial, Gear, and Taylor polynomial methods are chosen for analysis. To obtain accurate solutions, the time steps of the Hermite polynomial and Taylor polynomial methods are chosen to be h = 0.0001 s and h = 0.001 s, respectively, which are generally used as solution benchmarks [17][18][19]. The results for step reactivity ρ = 0.0015 and ρ = 0.0032 are presented in Tables 1 and 2.…”

Section: Numerical and Analytical Resultsmentioning

confidence: 99%

“…A small step results in a long computing time, and more importantly, there is large accumulated error due to *These authors contributed equally to this work †Corresponding author (email: Cwz2@21cn.com) the many computation steps. Many researchers have attempted to solve this problem and some relatively effective numerical methods have been proposed, such as the finite-difference method [5], finite-element method [6], Runge-Kutta procedure [7], quasistatic method [8,9], piecewise polynomial approach [10], singular perturbation method [11], stiffness confinement method [12], power series solution [13][14][15], and Padé approximation [16][17][18].…”

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confidence: 99%

An accurate solution of point kinetics equations of one-group delayed neutrons and an extraneous neutron source for step reactivity insertion

Xue-Li

Chen

2010

Chin. Sci. Bull.

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The continuous indication of the neutron density and its rate of change are important for the safe startup and operation of reactors. The best way to achieve this is to obtain analytical solutions of the neutron kinetics equations because none of the developed numerical methods can well satisfy the need for real-time or even super-time computation for the safe startup and operation of reactors in practice. In this paper, an accurate analytical solution of point kinetics equations with one-group delayed neutrons and an extraneous neutron source is proposed to calculate the change in neutron density, where the whole process from the subcritical stage to critical and supercritical stages is considered for step reactivity insertions. The accurate analytical solution can also be used as a benchmark of all numerical methods employed to solve stiff neutron kinetics equations. neutron kinetics, accurate solution, extraneous neutron source, step reactivity Citation:Li H F, Shang X L, Chen W Z. An accurate solution of point kinetics equations of one-group delayed neutrons and an extraneous neutron source for step reactivity insertion.Equations of point reactor neutron kinetics describe variations in the densities of neutrons and delayed neutron precursors, and are coupled but stiff equations. Huge disparities between the life-spans of the prompt neutron and delayed neutron lead to time constants with different orders of magnitude for the solutions of fast and slow components. It is well accepted that analytical solutions can be obtained only under simplified conditions, such as there being no extraneous neutron source, there being a prompt jump [1-3], and there being a constant neutron source [4]. These analytical solutions can only be used for qualitative rather than quantitative computation. For the latter, researchers have to turn to a numerical method. However, a very small step size has to be adopted for stability when applying a general numerical method such as the Euler, Runge-Kutta, or Adams method. A small step results in a long computing time, and more importantly, there is large accumulated error due to *These authors contributed equally to this work †Corresponding author (email: Cwz2@21cn.com) the many computation steps. Many researchers have attempted to solve this problem and some relatively effective numerical methods have been proposed, such as the finite-difference method [5], finite-element method [6], Runge-Kutta procedure [7], quasistatic method [8,9], piecewise polynomial approach [10], singular perturbation method [11], stiffness confinement method [12], power series solution [13-15], and Padé approximation [16-18]. Other methods [19] are being proposed and discussed.However, none of these methods can satisfy the need for real-time or even super-time computation in the safe operation of a reactor in practice. With so many numerical methods available, which should we choose? Can we in fact obtain an accurate analytical solution to the equations of point reactor neutron kinetics for step reactivity inse...

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“…And there are several methods especially adapted for solving the initial value problems for stiff systems of ordinary differential equations (Aboanber and Hamada, 2003;Aboanber, 2004;Tashakor et al, 2010). Among the methods are numerical integration using Simpson's rule, finite element method, Runge-Kutta procedures, quasi-static method, piecewise polynomial approach and other methods (Li et al, 2010;Abdallah and Nahla, 2011;Hamada, 2013).…”

Section: Introductionmentioning

confidence: 99%

Explicit appropriate basis function method for numerical solution of stiff systems

Chen

Xiao

et al. 2015

Annals of Nuclear Energy

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On pade′ approximations to the exponential function and application to the point kinetics equations

Cited by 33 publications

References 22 publications

Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

An accurate solution of point kinetics equations of one-group delayed neutrons and an extraneous neutron source for step reactivity insertion

Explicit appropriate basis function method for numerical solution of stiff systems

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