ON THE CONSTRUCTION OF EFFICIENT METHODS FOR SECOND ORDER IVPS WITH OSCILLATING SOLUTION

In this paper we describe procedures for the construction of efficient methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions. Based on the described procedures we develop two simple and efficient multistep methods for the solution of the above problems. The first method is exponentially-fitted and trigonometrically-fitted and the second has a minimal phase-lag. Both methods are symmetric. Numerical results obtained for several well known problems show the efficiency of the new methods when they are compared with known methods in the literature.

[1]  T. Simos.  High Algebraic Order Methods with Minimal Phase-Lag for Accurate Solution of the Schrödinger Equation , 1998 .

[2]  T. E. Simos,et al.  EXPONENTIALLY FITTED RUNGE–KUTTA FOURTH ALGEBRAIC ORDER METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS , 1999 .

[3]  José M. Ferrándiz,et al.  A General Procedure For the Adaptation of Multistep Algorithms to the Integration of Oscillatory Problems , 1998 .

[4]  Xiao-yan Liu,et al.  Numerical solution of one‐dimensional time‐independent Schrödinger equation by using symplectic schemes , 2000 .

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

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

[7]  Theodore E. Simos,et al.  A Family of Numerov-type Exponentially Fitted Methods for the Numerical Integration of the Schrödinger Equation , 1997, Comput. Chem..

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

[9]  T. E. Simos,et al.  A sixth-order exponentially fitted method for the numerical solution of the radial , 1990 .

[10]  T. E. Simos,et al.  SIMPLE AND ACCURATE EXPLICIT BESSEL AND NEUMANN FITTED METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION , 2000 .

[11]  T. E. Simos,et al.  A generator of high-order embedded P-stable methods for the numerical solution of the Schro¨dinger equation , 1996 .

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

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

[14]  T. E. Simos.  Predictor-corrector phase-fitted methods for Y{double_prime} = F(X,Y) and an application to the Schroedinger equation , 1995 .

[15]  Jesús Vigo-Aguiar,et al.  AN ADAPTED SYMPLECTIC INTEGRATOR FOR HAMILTONIAN PROBLEMS , 2001 .

[16]  John M. Blatt,et al.  Practical points concerning the solution of the Schrödinger equation , 1967 .

[17]  M. M. Chawla,et al.  Numerov made explicit has better stability , 1984 .

[18]  Tom Lyche,et al.  Chebyshevian multistep methods for ordinary differential equations , 1972 .

[19]  T. E. Simos,et al.  Embedded methods for the numerical solution of the Schrödinger equation , 1996 .

[20]  A Family Of Hybrid Eighth Order Methods With Minimal Phase-Lag For The Numerical Solution Of The Schrödinger Equation And Related Problems , 2000 .

[21]  S. Tremaine,et al.  Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits , 1990 .

[22]  A. C. Allison,et al.  The numerical solution of coupled differential equations arising from the Schrödinger equation , 1970 .

[23]  Theodore E. Simos,et al.  On Finite Difference Methods for the Solution of the Schrödinger Equation , 1999, Comput. Chem..

[24]  Tom E. Simos,et al.  An Eighth-Order Method With Minimal Phase-Lag For Accurate Computations For The Elastic Scattering Phase-Shift Problem , 1996 .

[25]  T. E. Simos.  NEW NUMEROV-TYPE METHODS FOR COMPUTING EIGENVALUES, RESONANCES, AND PHASE SHIFTS OF THE RADIAL SCHRODINGER EQUATION , 1997 .

[26]  A. D. Raptis,et al.  A variable step method for the numerical integration of the one-dimensional Schrödinger equation , 1985 .

[27]  T. E. Simos,et al.  Some new Numerov-type methods with minimal phase lag for the numerical integration of the radial Schrödinger equation , 1994 .

[28]  Moawwad E. A. El-Mikkawy,et al.  High-Order Embedded Runge-Kutta-Nystrom Formulae , 1987 .

[29]  T. Simos.  EIGHTH-ORDER METHOD FOR ACCURATE COMPUTATIONS FOR THE ELASTIC SCATTERING PHASE-SHIFT PROBLEM , 1998 .

[30]  T. E. Simos,et al.  A Family Of Numerov-Type Exponentially Fitted Predictor-Corrector Methods For The Numerical Integrat , 1996 .

[31]  A. C. Allison,et al.  Exponential-fitting methods for the numerical solution of the schrodinger equation , 1978 .

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

[33]  Tom E. Simos.  A New Numerov-Type Method For Computing Eigenvalues And Resonances Of The Radial Schrödinger Equation , 1996 .

[34]  T. Simos.  An Eighth Order Exponentially Fitted Method for the Numerical Solution of the Schrödinger Equation , 1998 .