Functionally-fitted block methods for second order ordinary differential equations

Abstract Functionally-fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions. In this paper, we propose a new type of functionally-fitted block methods for initial value problems of second order ordinary differential equations. The basic theory for the proposed methods is established. First, we present a sufficient condition for the existence of the functionally-fitted block methods. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the r -point functionally-fitted block method is convergent of order at least r for second order ordinary differential equations. Numerical experiments are conducted to demonstrate the validity and efficiency of the functionally-fitted block methods by comparison with some existing numerical methods.

[1]  Nguyen Huu Cong,et al.  Analysis of trigonometric implicit Runge-Kutta methods , 2007 .

[2]  N. S. Hoang.  Functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström methods , 2014, 1410.4090.

[3]  H. A. Watts,et al.  Block Implicit One-Step Methods* , 1969 .

[4]  Andrew G. Glen,et al.  APPL , 2001 .

[5]  Hongjiong Tian,et al.  Continuous block implicit hybrid one-step methods for ordinary and delay differential equations , 2011 .

[6]  H. A. Watts,et al.  A-stable block implicit one-step methods , 1972 .

[7]  Beatrice Paternoster,et al.  Computation of the Interval of Stability of Runge-Kutta-Nyström Methods , 1998, J. Symb. Comput..

[8]  John P. Coleman,et al.  Mixed collocation methods for y ′′ =f x,y , 2000 .

[9]  G. Vanden Berghe,et al.  Optimal implicit exponentially-fitted Runge–Kutta methods , 2001 .

[10]  Beatrice Paternoster,et al.  Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday , 2012, Comput. Phys. Commun..

[11]  Hongjiong Tian,et al.  Functionally-fitted block methods for ordinary differential equations , 2014, J. Comput. Appl. Math..

[12]  Simeon Ola Fatunla,et al.  Block methods for second order odes , 1991, Int. J. Comput. Math..

[13]  R. D'Ambrosio,et al.  Parameter estimation in exponentially fitted hybrid methods for second order differential problems , 2011, Journal of Mathematical Chemistry.

[14]  H. De Meyer,et al.  Weights of the exponential fitting multistep algorithms for first-order ODEs , 2001 .

[15]  Liviu Gr. Ixaru,et al.  P-stability and exponential-fitting methods for y″″ = f(x, y) , 1996 .

[16]  K. Ozawa.  A functional fitting Runge-Kutta-Nyström method with variable coefficients , 2002 .

[17]  Kazufumi Ozawa,et al.  A functional fitting Runge-Kutta method with variable coefficients , 2001 .

[18]  J. Lambert,et al.  Symmetric Multistip Methods for Periodic Initial Value Problems , 1976 .

[19]  T. E. Simos,et al.  An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions , 1998 .

[20]  Beatrice Paternoster,et al.  Two-step hybrid collocation methods for y"=f(x, y) , 2009, Appl. Math. Lett..

[21]  Samuel N. Jator,et al.  Block hybrid method using trigonometric basis for initial value problems with oscillating solutions , 2012, Numerical Algorithms.

[22]  Kazufumi Ozawa.  A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method , 2005 .

[23]  Beatrice Paternoster,et al.  A note on the capacitance matrix algorithm, substructuring, and mixed or Neumann boundary conditions , 1987 .

[24]  H. De Meyer,et al.  Frequency evaluation in exponential fitting multistep algorithms for ODEs , 2002 .

[25]  Nguyen Huu Cong,et al.  On functionally-fitted Runge–Kutta methods , 2006 .

[26]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[27]  Ch. Tsitouras.  EXPLICIT EIGHTH ORDER TWO-STEP METHODS WITH NINE STAGES FOR INTEGRATING OSCILLATORY PROBLEMS , 2006 .

[28]  J. M. Franco.  Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators , 2002 .

[29]  Raffaele D'Ambrosio,et al.  Exponentially fitted two-step hybrid methods for y″=f(x, y) , 2011, J. Comput. Appl. Math..

[30]  E. Hairer,et al.  A theory for Nyström methods , 1975 .

[31]  B. P. Sommeijer,et al.  Explicit, high-order Runge-Kutta-Nystro¨m methods for parallel computers , 1993 .

[32]  B. M. Fulk.  MATH , 1992 .

[33]  T. Simos.  Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation , 2010 .

[34]  P. Chartier.  L-stable parallel one-block methods for ordinary differential equations , 1994 .

[35]  Hans Van de Vyver,et al.  Frequency evaluation for exponentially fitted Runge-Kutta methods , 2005 .

[36]  D. G. Bettis,et al.  Stabilization of Cowell's method , 1969 .

[37]  A block method for the numerical integration of stiff systems of ordinary differential equations , 1979 .

[38]  Hongjiong Tian,et al.  Continuous Block Theta-Methods for Ordinary and Delay Differential Equations , 2009, SIAM J. Sci. Comput..

[39]  H. De Meyer,et al.  Exponentially fitted variable two-step BDF algorithm for first order ODEs☆ , 2003 .