An Algorithm to Optimize the Calculation of the Fourth Order Runge-Kutta Method Applied to the Numerical Integration of Kinetics Coupled Differential Equations — Oak Academic Publishing
Research ArticleOpen AccessGoogle Scholar indexed
An Algorithm to Optimize the Calculation of the Fourth Order Runge-Kutta Method Applied to the Numerical Integration of Kinetics Coupled Differential Equations
Institute of Physics, University of S?o Paulo, S?o Paulo, 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
1 Institute of Physics, University of S?o Paulo, S?o Paulo, Brazil
2 Institute of Physics, University of S?o Paulo, S?o Paulo, Brazil
3 Institute of Mathematics and Computer Science, University of S?o Paulo, S?o Carlos, Brazil
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.
E. I. Adirovitch, “The Formula of Becquerel and the Elementary Decay Law of Luminescence of Crystal Phosphors,” Journal of Physique Archives (Paris), Vol. 17, No. 8-9, 1956, pp. 705-707. doi:10.1051/jphysrad:01956001708-9070500
S. W. S. McKeever and R. Chen, “Luminescence Models,” Radiation Measurements, Vol. 27, No. 5-6, 1997, pp. 625-661. doi:10.1016/S1350-4487(97)00203-5
C. M. Sunta, W. E. F. Ayta, R. N. Kulkarni, T. M. Piters and S. Watanabe, “Theoretical Models of Thermoluminescence and Their Relevance in Experimental Work,” Radiation Protection Dosimetry, Vol. 84, No. 1-4, 1999, pp. 25-28. doi:10.1093/oxfordjournals.rpd.a032730
S. Isotani, W. W. Furtado, R. Antonini and O. L. Dias, “Line-Shape and Thermal Kinetics Analysis of the Fe2+-Band in Brazilian Green Beryl,” American Mineralogist, Vol. 74, No. 1, 1989, pp. 432-438.
R. Antonini, S. Isotani, W. W. Furtado, W. M. Pontuschka and S. R. Rabbani, “Study of the Decay Kinetics of Irradiation Induced Green Color in Brazilian Spodumene,” Proceedings of the Brazilian Academy of Sciences, Vol. 62, No. 1, 1990, pp. 39-43.
A. S. Ito and S. Isotani, “Heating Effects on the Optical Absorption Spectra of Irradiated, Natural Spodumene,” Radiation Effects and Defects in Solids: Incorporating Plasma Science and Plasma Technology, Vol. 116, No. 4, 1991, pp. 307-314. doi:10.1080/10420159108220737
S. Isotani, A. T. Fujii, R. Antonini, W. M. Pontuschka, S. R. Rabbani and W. W. Furtado, “Luminescence Study of Spodumene,” Proceedings of the Brazilian Academy of Sciences, Vol. 62, No. 2, 1990, pp. 107-113.
S. Isotani, W. W. Furtado, R. Antonini, A. R. Blak, W. M. Pontuschka, T. Tomé and S. R. Rabbani, “Decay Kinetics Study of Atomic Hydrogen in a-Si:(H,O,N) and Natural Beryl,” Physical Review B, Vol. 42, No. 10, 1990, pp. 5966-5972. doi:10.1103/PhysRevB.42.5966
M. B. Camargo, W. M. Pontuschka and S. Isotani, “Electron Paramagnetic Resonance of Atomic Hydrogen Centers in Rubellite,” Proceedings of the Brazilian Academy of Sciences, Vol. 59, No. 1, 1987, pp. 293-298.
A. R. Blak, W. M. Pontuschka and S. Isotani, “Electron Paramagnetic Resonance of Hydrogen Centers in Natural Beryl,” Proceedings of the Brazilian Academy of Sciences, Vol. 60, No. 1, 1988, pp. 9-12.
S. Sciortino, “Growth, Characterization and Properties of CVD Diamond Films for Applications as Radiation Detectors,” La Rivista del Nuovo Cimento, Vol. 22, No. 10, 1999, pp. 1-89. doi:10.1007/BF02872270
T. Budinas, A. Jankauskas, P. Mackup and A. Smilga, “Charge Transport Peculiarities in Thin-Layer Photodielectric System As2Se3Tlx-Sb2S3,” Thin Solid Films, Vol. 36, No. 2, 1976, pp. 277-280. doi:10.1016/0040-6090(76)90021-3
A. E. do Nascimento, P. Trzesniak, M. E. G. Valerio and J. F. de Lima, “On the Error in the Activation Energy Obtained by the Initial Raise Method for Thermally Stimulated Processes in Dielectrics,” Radiation Effects and Defects in Solids: Incorporating Plasma Science and Plasma Technology, Vol. 134, No. 1-4, 1995, pp. 147-152. doi:10.1080/10420159508227202
A. Opanowicz, “Interactive Kinetics in Thermally Stimulated Luminescence in Insulators and Semiconductors,” Physica B: Condensed Matter, Vol. 271, No. 1-4, 1999, pp. 273-283. doi:10.1016/S0921-4526(99)00208-2
L. Brizhik, F. Musumeci, A. Scordino and A. Triglia, “The Soliton Mechanism of the Delayed Luminescence of Biological Systems,” EPL (Europhysics Letters), Vol. 52, No. 2, 2000, pp. 238-244. doi:10.1209/epl/i2000-00429-5
R. Biswas and B.C. Pan, “Defect Kinetics in New Model of Metastability in a-Si:H,” Journal of Non-Crystalline Solids, Vol. 299-302, Part 1, 2002, pp. 507-510. doi:10.1016/S0022-3093(01)00961-9
A. D. Drozdov, J. De C. Christiansen, R. Klitkou and C. G. Potarniche, “Effect of Annealing on Viscoplasticity of Polymer Blends: Experiments and Modeling,” Computational Materials Science, Vol. 50, No. 1, 2010, pp. 59-64. doi:10.1016/j.commatsci.2010.07.007
W. W. Furtado, T. Tomé, S. Isotani, R. Antonini, A. R. Blak, W. M. Pontuschka and S. R. Rabbani, “Numerical Integration Method Applied to the Study of Atomic Hydrogen in Aluminoborate Glass,” Proceedings of the Brazilian Academy of Sciences, Vol. 61, No. 4, 1989, pp. 397-403.
A. Mizukami, S. Isotani, S. R. Rabbani and W. M. Pontuschka, “Approximate Solution for Kinetic Differential Equations,” Il Nuovo Cimento D, Vol. 15, No. 4, 1993, pp. 637-645. doi:10.1007/BF02482398
P. Kelly, M. J. Laubitz and P. Braunlich, “Exact Solutions of the Kinetic Equations Governing Thermally Stimulated Luminescence and Conductivity,” Physical Review B, Vol. 4, No. 6, 1971, pp. 1960-1968. doi:10.1103/PhysRevB.4.1960
J. C. Butcher, “Runge-Kutta Methods in Modern Computation, Part I: Fundamental Concepts,” Computers in Physics, Vol. 8, No. 4, 1994, pp. 411-415. doi:10.1063/1.168500
P. Rentrop, “Partitioned Runge-Kutta Methods with Stiffness Detection and Stepsize Control,” Numerische Mathematik, Vol. 47, No. 4, 1985, pp. 545-564. doi:10.1007/BF01389456
A. Brunetti, “A Fast and Precise Genetic Algorithm for a Non-Linear Fitting Problem,” Computer Physics Communications, Vol. 124, No. 2-3, 2000, pp. 204-211. doi:10.1016/S0010-4655(99)00454-3
F. Khorasheh, A. M. Ahmadi and A. Gerayeli, “Application of Direct Search Optimization for Pharmacokinetic Parameter Estimation,” Journal of Pharmacy and Pharmaceutical Science, Vol. 2, No. 3, 1999, pp. 92-98.
P. R. Bevington and D. K. Robinson, “Data Reduction and Error Analysis for the Physical Science,” 2nd Edition, McGraw-Hill, New York, 1992.
S. Isotani and A. T. Fujii, “A Grid Procedure Applied to the Determination of Parameters of a Kinetic Process,” Computer Physics Communications, Vol. 151, No. 1, 2003, pp. 1-7. doi:10.1016/S0010-4655(02)00692-6
W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Numerical Recipes in C++: The Art of Scientific Computation,” Cambridge University Press, New York, 1988.
D. W. Zingg and T. T. Chisholm, “Runge-Kutta Methods for Linear Ordinary Differential Equations,” Applied Numerical Mathematics, Vol. 31, No. 2, 1999, pp. 227-238. doi:10.1016/S0168-9274(98)00129-9
R. I. Mclachlan, Y. Sun and P. S. P. Tse, “Linear Stability of Partitioned Runge-Kutta Methods,” SIAM Journal on Numerical Analysis, Vol. 49, No. 1, 2011, pp. 232-263. doi:10.1137/100787234
G. Dimarco and L. Pareschi, “Exponential Runge-Kutta Methods for Stiff Kinetic Equations,” SIAM Journal on Numerical Analysis, Vol. 49, No. 5, 2011, pp. 2057-2077. doi:10.1137/100811052
J. Stoer and R. Bulirsch, “Introduction to Numerical Analysis,” Springer Verlag, New York, 1980.
C. W. Cear, “Numerical Initial Value Problems in Ordinary Differential Equations,” Prentice Hall, Upper Saddle River, 1971.
K. Gurski and S. O’Sullivan, “A Stability Study of a New Explicit Numerical Scheme for a System of Differential Equations with a Large Skew-Symmetric Component,” SIAM Journal on Numerical Analysis, Vol. 49, No. 1, 2011, pp. 368-386. doi:10.1137/090775804
M. J. Feigembaum, “Universal Behavior in Nonlinear Systems,” Physica D: Nonlinear Phenomena, Vol. 7, No. 1-3, 1983, pp. 16-39. doi:10.1016/0167-2789(83)90112-4
H. J. L. Hagebeuk and P. Kivits, “Determination of Trapping Parameters from the Conventional Model for Thermally Stimulated Luminescence and Conductivity,” Physica B+C, Vol. 83, No. 3, 1976, pp. 289-294. doi:10.1016/0378-4363(76)90123-6
A. E. Aboanber, “Stability of Generalized Runge-Kutta Methods for Stiff Kinetics Coupled Differential Equations,” Journal of Physics A: Mathematical and General, Vol. 39, No. 8, 2006, pp. 1859-1876. doi:10.1088/0305-4470/39/8/006
X. J. Wang and S. Q. Gan, “A Runge-Kutta Type Scheme for Nonlinear Stochastic Partial Differential Equations with Multiplicative Trace Class Noise,” Numerical Algorithms, 2012. (To Be Published). doi:10.1007/s11075-012-9568-8
H. Yuan, J. Zhao and Y. Xu, “Some Stability and Convergence of Additive Runge-Kutta Methods for Delay Differential Equations with Many Delays,” Journal of Applied Mathematics, Vol. 2012, No. 1, 2012, pp. 1-18. doi:10.1155/2012/456814
J. S. C. Prentice, “Stepsize Selection in Explicit RungeKutta Methods for Moderately Stiff Problems,” Applied Mathematics, Vol. 2, No. 6, 2011, pp. 711-717. doi:10.4236/am.2011.26094
M. N. Spijker, “Stepsize Conditions for General Monotonicity in Numerical Initial Value Problems,” SIAM Journal on Numerical Analysis, Vol. 45, No. 3, 2007, pp. 1226-1245. doi:10.1137/060661739
S. Gottlieb, D. I. Ketcheson and C.-W. Shu, “Strong Stability Preservation Runge-Kutta and Multistep Time Discretizations,” World Scientific Publishing Co., Singapore, 2011. doi:10.1142/7498
A. Valinejad and S. M. Hosseini, “A Stepsize Control Algorithms for SDEs with Small Noise Based on Stochastic Runge-Kutta Maruyama Methods,” Numerical Algorithms, Vol. 61, No. 3, 2012, pp. 479-498. doi:10.1007/s11075-012-9544-3
W. M. Pontuschka and S. Isotani, “A Model for the Stabilization of Atomic Hydrogen Centers in Borate Glasses,” Physica B: Condensed Matter, Vol. 404, No. 21, 2009, pp. 4312-4315. doi:10.1016/j.physb.2009.08.032
D. I. Ketcheson and A. J. Ahmadia, “Optimal RungeKutta Stability Regions,” 2012. http://arxiv.org/pdf/1201.3035v2.pdf