Phase-fitted and amplification-fitted two-step hybrid methods for y˝= f ( x,y )

This paper provides a theoretical framework for a new type of phase-fitted and amplification-fitted two-step hybrid (FTSH) methods which is introduced by the author in [H. Van de Vyver, A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems, Internat. J. Modern Phys. C 17 (2006) 663-675]. The methods constitute a modification of dissipative two-step hybrid methods in the sense that two free parameters are added to eliminate the phase-lag and the amplification error. The methods are useful only when a good estimate of the frequency of the problem is known in advance. The parameters depend on the product of the estimated frequency and the stepsize. The algebraic order, zero-stability, stability and phase properties are examined. The theory is illustrated with sixth-order explicit FTSH methods. Numerical results carried out on an assortment of test problems show the relevance of the theory.

[1]  G. Avdelas,et al.  Dissipative high phase-lag order numerov-type methods for the numerical solution of the Schrodinger equation , 2000 .

[2]  A. D. Raptis,et al.  A four-step phase-fitted method for the numerical integration of second order initial-value problems , 1991 .

[3]  R. Van Dooren.  Stabilization of Cowell's classical finite difference method for numerical integration , 1974 .

[4]  T. E. Simos,et al.  Controlling the error growth in long–term numerical integration of perturbed oscillations in one or several frequencies , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[5]  L.Gr. Ixaru,et al.  Numerov method maximally adapted to the Schro¨dinger equation , 1987 .

[6]  Hans Van de Vyver,et al.  An embedded phase-fitted modified Runge–Kutta method for the numerical integration of the radial Schrödinger equation , 2006 .

[7]  Hans Van de Vyver,et al.  Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems , 2005, Comput. Phys. Commun..

[8]  Ch. Tsitouras,et al.  Explicit Numerov Type Methods with Reduced Number of Stages , 2003 .

[9]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

[10]  Hans Van de Vyver.  A fourth-order symplectic exponentially fitted integrator , 2006, Comput. Phys. Commun..

[11]  Theodore E. Simos.  Dissipative High Phase-lag Order Numerov-type Methods for the Numerical Solution of the Schrödinger Equation , 1999, Comput. Chem..

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

[13]  Hans Van de Vyver,et al.  On the generation of P-stable exponentially fitted Runge-Kutta-Nyström methods by exponentially fitted Runge-Kutta methods , 2006 .

[14]  J. M. Franco.  A class of explicit two-step hybrid methods for second-order IVPs , 2006 .

[15]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[16]  John P. Coleman,et al.  Order conditions for a class of two‐step methods for y″ = f (x, y) , 2003 .

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

[18]  Hans Van de Vyver.  A PHASE-FITTED AND AMPLIFICATION-FITTED EXPLICIT TWO-STEP HYBRID METHOD FOR SECOND-ORDER PERIODIC INITIAL VALUE PROBLEMS , 2006 .

[19]  Zacharias A. Anastassi,et al.  A dispersive-fitted and dissipative-fitted explicit Runge–Kutta method for the numerical solution of orbital problems , 2004 .

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

[21]  J. M. Franco,et al.  High-order P-stable multistep methods , 1990 .

[22]  M H Chawla,et al.  A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value , 1986 .

[24]  Tom E. Simos,et al.  A Numerov-type method for the numerical solution of the radial Schro¨dinger equation , 1991 .

[25]  K. Ozawa.  A four-stage implicit Runge-Kutta-Nyström method with variable coefficients for solving periodic initial value problems , 1999 .

[26]  L. Brusa,et al.  A one‐step method for direct integration of structural dynamic equations , 1980 .

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

[28]  Zacharias A. Anastassi,et al.  Special Optimized Runge-Kutta Methods for IVPs with Oscillating Solutions , 2004 .

[29]  Theodore E. Simos,et al.  A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problem , 1991, Int. J. Comput. Math..

[30]  Beatrice Paternoster,et al.  Stability regions of one step mixed collocation methods for y″=f(x,y) , 2005 .

[31]  M. M. Chawla,et al.  Intervals of periodicity and absolute stability of explicit nyström methods fory″=f(x,y) , 1981 .

[32]  Ch. Tsitouras,et al.  Phase-fitted Numerov type methods , 2007, Appl. Math. Comput..

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

[34]  M. M. Chawla,et al.  An explicit sixth-order method with phase-lag of order eight for y ″= f ( t , y ) , 1987 .

[35]  C. Tsitouras.  Explicit two-step methods for second-order linear IVPs , 2002 .