Construcción y estudio de códigos adaptativos de linealización local para ecuaciones diferenciales ordinarias

• Sotolongo Aguiar, Alina ^[1] ; Jiménez Sobrino, Juan Carlos ^[2]
  1. [1] Universidad de La Habana

     Universidad de La Habana

     Cuba

     
  2. [2] Instituto de Cibernética Matemática y Física

     Instituto de Cibernética Matemática y Física

     Cuba

     
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 21, Nº. 1, 2014, págs. 21-53
• Idioma: español
• DOI: 10.15517/rmta.v21i1.14136
• Títulos paralelos:
  • Construction and study of local linearization adaptive codes for ordinary differential equations
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    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.

  • English

    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. 

• Referencias bibliográficas
  • Beyn, W.J. (1987) “On the numerical approximation of phase portraits near stationary points”, SIAM J. Numer. Anal. 24: 1095–1113.
  • Carbonell, F.; Jimenez, J.C.; Biscay, R.J. (2006) “Weak local linear discretizations for stochastic differential equations: convergence and...
  • Carbonell, F.; Jimenez, J.C.; Biscay, R.; de la Cruz, H. (2005) “The Local Linearization method for numerical integration of random differential...
  • Cartwright, J.H.E.; Piro, O. (1992) “The dynamics of Runge-Kutta methods”, Int. J. Bifurc. Chaos 2: 427–449.
  • Cox, S.M.; Matthews, P.C. (2002) “Exponential time differencing for stiff systems”, J. Comput. Phys. 176: 430–455.
  • de la Cruz, H.; Biscay, R.J.; Carbonell, F.; Jimenez, J.C.; Ozaki, T. (2006) “Local Linearization-Runge Kutta (LLRK) methods for solving ordinary...
  • de la Cruz, H.; Biscay, R.J.; Carbonell, F.; Ozaki, T.; Jimenez, J.C. (2007) A higher order Local Linearization method for solving ordinary...
  • de la Cruz, H.; Biscay, R.J.; Jimenez, J.C.; Carbonell, F.; Ozaki, T. (2010) “High order local linearization methods: an approach for constructing...
  • 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...
  • Hairer, E.; Norsett, S.P.; Wanner, G. (1993) Solving Ordinary Differential Equations I, 2nd ed. Springer-Verlag, Berlin.
  • Hairer, E.; Wanner, G. (1996) Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 3th ed. Springer-Verlag,...
  • Hochbruck, M.; Ostermann, A.; Schweitzer, J. (2009) “Exponential Rosenbrock type methods”, SIAM J. Numer. Anal. 47: 786–803.
  • Hochbruck, M.; Ostermann, A. (2010) “Exponential integrators”, Acta Numer. 19: 209–286.
  • Iserles, A. (1990) “Stability and dynamics of numerical methods for non-linear ordinary differential equations”, IMA J. Numer. Anal. 10: 1–30.
  • Jimenez, J.C.; Biscay, R.; Mora, C.; Rodriguez, L.M. (2002) “Dynamic properties of the Local Linearization method for initial-value problems”,...
  • Jimenez, J.C.; Carbonell, F. (2005) “Rate of convergence of local linearization schemes for initial-value problems”, Appl. Math. Comput. 171:...
  • Jimenez, J.C.; Pedroso, L.; Carbonell, F.; Hernandez, V. (2006) “Local linearization method for numerical integration of delay differential...
  • 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
  • Prenter P.M. (1975) Splines and Variational Methods. John Wiley & Sons, New York.
  • Rahunanthana, A.; Stanescu, D. (2010) “High-order W-methods”, Journal of Computational and Applied Mathematics 233: 1798–1811.
  • Shampine, L.F.; Reichelt, M.W. (1997) “The MATLAB ODE Suite” SIAM Journal on Scientific Computing 18: 1–22.
  • Skufca, J.D. (2004) “Analysis still matters: a surprising instance of failure of Runge-Kutta-Felberg ODE solvers”, SIAM Review 46: 729–737
  • Stewart, I. (1992) “Numerical methods: Warning-handle with care!”, Nature 355: 16–17.
  • Van Loan, C.F.; Moler, C. (2003) “Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later”, SIAM Review 45:...