El propósito de este trabajo es construir códigos adaptativos del método de Linealización Local para Ecuaciones Diferenciales Ordinarias (EDO) y analizar su comportamiento numérico. Además, se estudia el efecto que sobre las propiedades de los códigos produce la variación en la precisión de las aproximaciones de Padé utilizadas.
Palabras clave: Integradores numéricos, código adaptativo, método de linealización local, método de Runge Kutta, fórmula de Padé.
Abstract
The aim of this work is to construct adaptive integrators for ordinary differential equations based on the Local Linearization method. Different orders of the involved Padé approximation are considered and their effect on the adaptive integrators is studied. ]]>
Keywords: numerical integrators, adaptive codes, local linearization method, Runge–Kutta method, Padé aproximation.
Mathematics Subject Classification: 65L05, 37M05, 65L06. Ver contenido disponible en pdf
Referencias
[1] Beyn,W.J. (1987) “On the numerical approximation of phase portraits near stationary points”, SIAM J. Numer. Anal. 24: 1095–1113. [ Links ]
[2] Carbonell, F.; Jimenez, J.C.; Biscay, R.J. (2006) “Weak local linear discretizations for stochastic differential equations: convergence and numerical schemes”, J. Comput. Appl. Math. 197: 578–596. [ Links ]
[3] Carbonell, F.; Jimenez, J.C.; Biscay, R.; de la Cruz, H. (2005) “The Local Linearization method for numerical integration of random differential equations”, BIT Numerical Mathematics 45: 1–14. [ Links ]
[4] Cartwright, J.H.E.; Piro, O. (1992) “The dynamics of Runge-Kutta methods”, Int. J. Bifurc. Chaos 2: 427–449. [ Links ] ]]>
[5] Cox, S.M.; Matthews, P.C. (2002) “Exponential time differencing for stiff systems”, J. Comput. Phys. 176: 430–455. [ Links ]
[6] de la Cruz, H.; Biscay, R.J.; Carbonell, F.; Jimenez, J.C.; Ozaki, T. (2006) “Local Linearization-Runge Kutta (LLRK) methods for solving ordinary differential equations”, in: Lecture Notes in Computer Sciences 3991, Springer-Verlag, Berlin: 132–139. [ Links ]
[7] de la Cruz, H.; Biscay, R.J.; Carbonell, F.; Ozaki, T.; Jimenez, J.C. (2007) A higherorder LocalLinearizationmethod for solvingordinary differential equations, Appl. Math. Comput. 185: 197–212. [ Links ] ]]>
[8] de la Cruz, H.; Biscay, R.J.; Jimenez, J.C.; Carbonell, F.; Ozaki, T. (2010) “High order local linearization methods: an approach for constructing Astable high order explicit schemes for stochastic differential equations with additive noise”, BIT Numerical Mathematics 50: 509–539. [ Links ]
[9] de la Cruz H., Biscay R.J., Jimenez J.C. and Carbonell F., Local Linearization-Runge-Kutta methods: A class of A-stable explicit integrators for dynamical systems, Math. Comput. Modell., 57 (2013) 720–740. [ Links ]
[10] Hairer, E.; Norsett, S.P.; Wanner, G. (1993) Solving Ordinary Differential Equations I, 2nd ed. Springer-Verlag, Berlin. [ Links ]
[11] Hairer, E.; Wanner, G. (1996) Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 3th ed. Springer-Verlag, Berlin. [ Links ]
[12] Hochbruck,M.; Ostermann, A.; Schweitzer, J. (2009) “Exponential Rosenbrock type methods”, SIAM J. Numer. Anal. 47: 786–803. [ Links ]
[14] Iserles, A. (1990) “Stability and dynamics of numerical methods for nonlinear ordinary differential equations”, IMA J. Numer. Anal. 10: 1–30. [ Links ] ]]>
[15] Jimenez, J.C.; Biscay, R.; Mora, C.; Rodriguez, L.M. (2002) “Dynamic properties of the Local Linearization method for initial-value problems”, Appl. Math.Comput. 126: 63–81. [ Links ]
[16] Jimenez, J.C.; Carbonell, F. (2005) “Rate of convergence of local linearization schemes for initial-value problems”, Appl. Math. Comput. 171: 1282–1295. [ Links ]
[17] Jimenez, J.C.; Pedroso, L.; Carbonell, F.; Hernandez, V. (2006) “Local linearization method for numerical integration of delay differential equations", SIAM J. Numer. Analysis 44: 2584–2609. [ Links ] ]]>
[18] Jimenez, J.C.; Sotolongo, A.; Sanchez-Bornot, J.M. (2012) Locally Linearized Runge Kutta method of Dormand and Prince, in: http://arxiv.org/abs/1209.1415.pdf [ Links ]
[19] Prenter P.M. (1975) Splines and Variational Methods. John Wiley & Sons, New York. [ Links ]
[20] Rahunanthana, A.; Stanescu, D. (2010) “High-order W-methods”, Journal of Computational and Applied Mathematics 233: 1798–1811. [ Links ]
[21] Shampine, L.F.; Reichelt, M.W. (1997) “The MATLAB ODE Suite” SIAM Journal on Scientific Computing 18: 1–22. [ Links ] ]]>
[22] Skufca, J.D. (2004) “Analysis still matters: a surprising instance of failure of Runge-Kutta-Felberg ODE solvers”, SIAM Review 46: 729–737 [ Links ]
[23] Stewart, I. (1992) “Numerical methods: Warning-handle with care!”, Nature 355: 16–17. [ Links ]
[24] Van Loan, C.F.; Moler, C. (2003) “Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later”, SIAM Review 45: 3–49. [ Links ]
*Correspondencia a: ]]>
Alina Sotolongo Aguiar:Facultad de Matemática y Computación, Universidad de la Habana, La Habana, Cuba.E-Mail: alina.sotolongo@gmail.com Juan Carlos Jiménez Sobrino: Instituto de Cibernética, Matemática y Física, Calle 15, No 551, La Habana, Cuba. E-Mail:jcarlos@icimaf.cu *Facultad de Matemática y Computación, Universidad de la Habana, La Habana, Cuba.E-Mail: alina.sotolongo@gmail.com †Instituto de Cibernética, Matemática y Física, Calle 15, No 551, La Habana, Cuba. E-Mail:jcarlos@icimaf.cu