P-stability‎, ‎TF and VSDPL technique in Obrechkoff methods for the numerical solution of the Schrodinger equation

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Faculty of Basic Science‎, ‎University‎ ‎of Maragheh‎, ‎P.O‎. ‎Box 55181-83111‎, ‎Maragheh‎, ‎Iran.

Abstract

Many simulation algorithms (chemical reaction systems, differential systems arising from the modeling of transient behavior in the process industries and etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta technique are used. For the simulation of chemical procedures the radial Schrodinger equation is used frequently. In the present paper we will study a symmetric two-step Obrechkoff method, in which we will use of technique of VSDPL (vanished some of derivatives of
phase-lag), for the numerical integration of the one-dimensional Schrodinger equation. We will show superiority of new method in stability, accuracy and efficiency. So we present a stability analysis and an error analysis based on the radial Schrodinger equation. Also we will apply the new proposed method to the resonance problem of the radial Schrodinger equation.

Keywords

Main Subjects


S. D. Achar, Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations, J. Appl. Math. Comput. 218 (2011), no. 5, 2237--2248.

U. A. Ananthakrishnaiah, P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems, Math. Comput. 49 (1987), no. 180, 553--559.

M. M. Chawla and P. S. Rao, A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial value problems, II, Explicit method, J. Comput. Appl. Math. 15 (1986), no. 3, 329--337.

M. M. Chawla, P. S. Rao and B. Neta, Two-step fourth order P-stable methods with phase-lag of order six for y”= f(t; y), J. Comput. Appl. Math. 16 (1986), no. 2, 233--236.

J. P. Coleman and L. Gr. Ixaru, P-stability and exponential-fitting methods for y=f(x, y), IMA J. Numer. Anal. 16 (1995), no. 2, 179--199.

L. G. Ixaru and M. Rizea, Comparison of some four-step methods for the numerical solution of the Schrodinger equation, Comput. Phys. Commun. 38 (1985), no. 3, 329--337.

L. G. Ixaru and M. Rizea, A Numerov-like scheme for the numerical solution of the Schrodinger equation in the deep continuum spectrum of energies, Comput. Phys. Commun. 19 (1980), 23--27.

J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976), no. 2, 189--202.

A. D. Raptis, Exponentially-fitted solutions of the eigenvalue Schrdinger equation with automatic error control, Comput. Phys. Comm. 28 (1983), no. 4, 427--431.

D. P. Sakas and T. E. Simos, Trigonometrically-fitted multiderivative methods for the numerical solution of the radial Schrodinger equation, Commun. Math. Comput. Chem. 53 (2005), no. 2, 299--320.

G. Saldanha and S. D. Achar, Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations, Appl. Math. Comput. 218 (2011), no. 5, 2237--2248.

T. E. Simos, A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial value problems, Proc. Roy. Soc. London Ser. A 441 (1993), no. 1912, 283--289.

A. Shokri, An explicit trigonometrically fitted ten-step method with phase-lag of order infinity for the numerical solution of radial Schrodinger equation, Appl. Comput. Math. 14 (2015), no. 1, 63--74.

A. Shokri, The symmetric two-step P-stable nonlinear predictor-corrector methods for the numerical solution of second order initial value problems, Bull. Iranian Math. Soc. 41 (2015), no. 1, 201--215.

A. Shokri and H. Saadat, High phase-lag order trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems, Numer. Algorithms 68 (2015), no. 2, 337--354.

A. Shokri and H. Saadat, Trigonometrically fitted high-order predictor-corrector method with phase-lag of order infinity for the numerical solution of radial Schrodinger equation, J. Math. Chem. 52 (2014), no. 7, 1870--1894.

M. Van Daele and G. Vanden Berghe, P-stable exponentially fitted Obrechkoff methods of arbitrary order for second order differential equations, Numer. Algorithms 46 (2007), no. 4, 333--350.

G. Vanden Berghe and M. Van Daele, Exponentially-fitted Obrechkoff methods for second-order differential equations, Applied Numerical Mathematics 59 (2009), no. 3-4, 815--829.

Z. Wang, D. Zhao, Y. Dai and D. Wu, An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial value problems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2058, 1639--1658.