An Algorithm to Optimize the Calculation of the Fourth Order Runge-Kutta Method Applied to the Numerical Integration of Kinetics Coupled Differential Equations

Affiliation(s)

Institute of Physics, University of S?o Paulo, S?o Paulo, Brazil.

Institute of Mathematics and Computer Science, University of S?o Paulo, S?o Carlos, Brazil.

Institute of Physics, University of S?o Paulo, S?o Paulo, Brazil.

Institute of Mathematics and Computer Science, University of S?o Paulo, S?o Carlos, Brazil.

Abstract

The kinetic electron trapping process in a shallow defect state and its subsequent thermal- or photo-stimulated promotion to a conduction band, followed by recombination in another defect, was described by Adirovitch using coupled rate differential equations. The solution for these equations has been frequently computed using the Runge-Kutta method. In this research, we empirically demonstrated that using the Runge-Kutta Fourth Order method may lead to incorrect and ramified results if the numbers of steps to achieve the solutions is not “large enough”. Taking into account these results, we conducted numerical analysis and experiments to develop an algorithm that determines the smallest non-critical number of steps in an automatic way to optimize the application of the Runge-Kutta Fourth Order method. This algorithm was implemented and tested in a variety of situations and the results have shown that our solution is robust in dealing with different equations and parameters.

The kinetic electron trapping process in a shallow defect state and its subsequent thermal- or photo-stimulated promotion to a conduction band, followed by recombination in another defect, was described by Adirovitch using coupled rate differential equations. The solution for these equations has been frequently computed using the Runge-Kutta method. In this research, we empirically demonstrated that using the Runge-Kutta Fourth Order method may lead to incorrect and ramified results if the numbers of steps to achieve the solutions is not “large enough”. Taking into account these results, we conducted numerical analysis and experiments to develop an algorithm that determines the smallest non-critical number of steps in an automatic way to optimize the application of the Runge-Kutta Fourth Order method. This algorithm was implemented and tested in a variety of situations and the results have shown that our solution is robust in dealing with different equations and parameters.

Cite this paper

S. Isotani, W. Pontuschka and S. Isotani, "An Algorithm to Optimize the Calculation of the Fourth Order Runge-Kutta Method Applied to the Numerical Integration of Kinetics Coupled Differential Equations,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1583-1592. doi: 10.4236/am.2012.311218.

S. Isotani, W. Pontuschka and S. Isotani, "An Algorithm to Optimize the Calculation of the Fourth Order Runge-Kutta Method Applied to the Numerical Integration of Kinetics Coupled Differential Equations,"

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http://arxiv.org/pdf/1201.3035v2.pdf