A highly accurate technique for the solution of the non-linear point kinetics equations

Abstract

The method of Taylor series expansion is used to develop a numerical solution to the reactor point kinetics equations. It is shown that taking a first order expansion of the neutron density and precursor concentrations at each time step gives results that are comparable to those obtained using other popular yet more complicated methods. The algorithm developed using a Taylor series expansion is simple, completely transparent, and highly accurate. The procedure is tested using a variety of initial conditions and input data, including step reactivity, ramp reactivity, sinusoidal, pulse, and zigzag reactivity. These results are compared to those obtained using other methods.

Neutron Density Calculation Using the Generalised Adams-Bashforth-Moulton Method

“…In the case of very fast ramp reactivities, the ABM method can be applied to a fast reactor with the parameters obtained in Picca [13] using the EPCA method, and from Yun Cai et al [22] using the ME3 and PCA/ME2 methods. The values of the kinetic parameters are: Table 7 shows the results obtained with the ABM method and those reported in the literature for a reactivity ρ = 1 ($)/s.…”

“…The Taylor series (TSM) was used to calculate the nuclear density with feedback reactivity [9], and the power series method (PWS) has been used to obtain approximate solutions with and without feedback [10][11]. [12] introduced a highly accurate algorithm, combining the Backward Euler Finite Difference method (BEFD) and [13] described a semi-analytical method to solve the equations of point kinetics with a technique called (EPCA) which iteratively corrects the error in the source term with good accuracy. The reduced form of the differential transform method (reduced DT Method) [14] was used to obtain the total neutron density.…”

A New Accurate Numerical Method Based on Shifted Chebyshev Series for Nuclear Reactor Dynamical Systems

“…A ramp reactivity input, ( ) = 0.1 , is considered with kinetic parameters listed in Table 1. Table 3 shows a comparison between our tenthorder SCS method, stiffness confinement method SCM, −weighting method [21], generalized Runge-Kutta method GRK [22], the analytical inversion method AIM [23], the piecewise constant approximation PCA [24], the numerical algorithm CORE [25], the better basis function BBF, Hermit polynomial methods [26], the generalized analytical exponential method, GAEM and Pade' approximation with treatment of inhour equation root [27], the efficient technique ET [28], the power series method PWS [10], the converged accelerated Taylor series CATS [12], the accurate solution [29], the enhancement of the piecewise constant approximation EPCA [8], the fundamental backward Euler difference scheme BEFD [13], the integral formulation Taylor series expansions ITS2 [2], the Haar wavelet operational method HWOM [30], the modified exponential time differencing ETD [7], the trigonometric Fourier-series solutions TFS [6], and the treatment theta method TTM [31]. Thus, the proposed method is compared with twenty of different numerical methods in Table 3.…”

“…Two values of neutron generation times are considered: the first is Λ = 10 −6 for fast reactor I and the second is Λ = 10 −8 for fast reactor II. In Table 4(a), the results of our SCS method, using step size ℎ = 0.0001, are compared with those of CATS and EPCA methods [8]. Up to = , the results of the SCS and CATS are completely coinciding while EPCA method differ in the last digit at = 50 .…”

“…The modified exponential time differencing method [7] is applied to solve the PKEs. Picca [8] provided a method based on piecewise constant approximation to solve the nonlinear PKEs. Wollmann da Silva [9] eliminated the stiffness character by splitting the corresponding matrix into a diagonal matrix plus a matrix that contains the remaining terms.…”