Generalized trigonometric Fourier-series method with automatic time step control for non-linear point kinetics equations
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Generalized trigonometric Fourier-series method with automatic time step control for non-linear point kinetics equations Yasser Mohamed Hamada1
Received: 12 March 2016 / Revised: 24 May 2017 / Accepted: 9 June 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract The trigonometric Fourier-series method (TFS) is generalized to provide approximate solutions for non-linear point kinetics equations with feedback using varying step sizes. This method can provide a very stable solution against the size of the discrete time step allowing much larger step sizes to be used. Systems of the point kinetics equations are solved using Fourier-series expansion over a partition of the total time interval. The approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high-order derivatives of the solution vector at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. This system is similar to the Vandermonde system. Two successive orders of the partial sums are used to estimate the local truncation error. This error and some other constrains are used to produce the largest step size allowable at each step while keeping the error within a specific tolerance. The process of calculating suitable step sizes should be automatic and inexpensive. Convergence and stability of the proposed method are discussed and a new formula is introduced to maintain stability. The proposed method solves the general linear and non-linear kinetics problems. The method has been applied to five different types of reactivities including step/ramp insertions with temperature feedback. The method is seemed to be valid for larger time intervals than those used in the conventional numerical integration, and is thus useful in reducing computing time. Computational results are found to be consistent with the analysis, they demonstrate that the convergence of the iteration scheme can be accelerated and the resulting computing times can be greatly reduced while maintain computational accuracy. Keywords Fourier series method · Reactor dynamics · Feedback · Step size control · Numerical solutions
Communicated by Jose Alberto Cuminato.
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Yasser Mohamed Hamada [email protected]; [email protected] Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt
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Y. M. Hamada
Mathematics Subject Classification 65G20
1 Introduction Numerical techniques for solving differential equations can be divided into two categories. Finite difference techniques, whereas the unknown function u(t) is represented by a table of numbers {y(t0 ), y(t1 ), . . . , y(tn )} approximating their values at a set of discrete points {t0 , t1 , . . . , tn }, so that u(ti ) ≈ y(ti ). The second category represents spectral methods, where the function u(t) can be represented by a set of orthogonal functions ∅n (t) over a specific interval
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