A family of P-stable exponentially‐fitted methods for the numerical solution of the Schrödinger equation

Journal of Mathematical Chemistry - Tập 25 - Trang 65-84 - 1999
T.E. Simos1
1Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, Xanthi, Greece

Tóm tắt

A family of P‐stable exponentially‐fitted methods for the numerical solution of the Schrödinger equation is developed in this paper. An application to the resonance problem of the radial Schrödinger equation indicates that the new method is generally more efficient than the previously developed exponentially‐fitted methods of the same kind.

Tài liệu tham khảo

J.M. Blatt, Practical points concerning the solution of the Schr¨odinger equation, J. Comput. Phys. 1 (1967) 382–396. J.R. Cash and A.D. Raptis, A high order method for the numerical solution of the one-dimensional Schr¨odinger equation, Comput. Phys. Comm. 33 (1984) 299–304. J.R. Cash, A.D. Raptis and T.E. Simos, A sixth-order exponentially fitted method for the numerical solution of the radial Schr¨odinger equation, J. Comput. Phys. 91 (1990) 413–423. M.M. Chawla and P.S. Rao, A Numerov-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) 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 y00 = f(x, y), J. Comput. Appl. Math. 16 (1986) 233–236. J.P. Coleman, Numerical methods for y00 = f(x, y) via rational approximation for the cosine, IMA J. Numer. Anal. 9 (1989) 145–165. J.P. Coleman and L.Gr. Ixaru, P-stability and exponential-fitting methods for y00 = f(x, y), IMA J. Numer. Anal. 16 (1996) 179–199. J.W. Cooley, An improved eigenvalue corrector formula for solving Schr¨odinger's equation for central fields, Math. Comp. 15 (1961) 363–374. J.R. Dormand, M.E. El-Mikkawy and P.J. Prince, Families of Runge-Kutta-Nystr¨om formulae, IMA J. Numer. Anal. 7 (1987) 423–430. J.R. Dormand and P.J. Prince, Runge-Kutta-Nystr¨om triples, Comput. Math. Appl. 13 (1987) 937–949. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962). G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, Toronto, 1950). L.Gr. Ixaru and M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978). L.Gr. Ixaru and M. Rizea, A Numerov-like scheme for the numerical solution of the Schr¨odinger equation in the deep continuum spectrum of energies, Comput. Phys. Comm. 19 (1980) 23–27. J. Killingbeck, Shooting methods for the Schr¨odinger equation, J. Phys. A: Math. Gen. 20 (1987) 1411–1417. H. Kobeissi and M. Kobeissi, On testing difference equations for the diatomic eigenvalue problem, J. Comput. Chem. 9 (1988) 844–850. H. Kobeissi and M. Kobeissi, A new variable step method for the numerical integration of the one-dimensional Schr¨odinger equation, J. Comput. Phys. 77 (1988) 501–512. H. Kobeissi, M. Kobeissi and A. El-Hajj, On computing eigenvalues of the Schr¨odinger equation for symmetrical potentials, J. Phys. A: Math. Gen. 22 (1989) 287–295. G.J. Kroes, The royal road to an energy-conserving predictor-corrector method, Comput. Phys. Comm. 70 (1992) 41–52. J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976) 189–202. L.D. Landau and F.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965). T. Lyche, Chebyshevian multistep methods for ordinary differential equations, Numer. Math. 19 (1972) 65–75. A.D. Raptis, On the numerical solution of the Schr¨odinger equation, Comput. Phys. Comm. 24 (1981) 1–4. A.D. Raptis, Two-step methods for the numerical solution of the Schr¨odinger equation, Computing 28 (1982) 373–378. A.D. Raptis, Exponentially-fitted solutions of the eigenvalue Schr¨odinger equation with automatic error control, Comput. Phys. Comm. 28 (1983) 427–431. A.D. Raptis, Exponential multistep methods for ordinary differential equations, Bull. Greek Math. Soc. 25 (1984) 113–126. A.D. Raptis and A.C. Allison, Exponential-fitting methods for the numerical solution of the Schr¨odinger equation, Comput. Phys. Comm. 14 (1978) 1–5. A.D. Raptis and J.R. Cash, Exponential and Bessel fitting methods for the numerical solution of the Schr¨odinger equation, Comput. Phys. Comm. 44 (1987) 95–103. T.E. Simos, Numerical solution of ordinary differential equations with periodical solution, Doctoral dissertation, National Technical University of Athens (1990). T.E. Simos, A four-step method for the numerical solution of the Schr¨odinger equation, J. Comput. Appl. Math. 30 (1990) 251–255. T.E. Simos, Some new four-step exponential-fitting methods for the numerical solution of the radial Schr¨odinger equation, IMA J. Numer. Anal. 11 (1991) 347–356. T.E. Simos, Exponential fitted methods for the numerical integration of the Schr¨odinger equation, Comput. Phys. Comm. 71 (1992) 32–38. T.E. Simos, Error analysis of exponential-fitted methods for the numerical solution of the onedimensional Schr¨odinger equation, Phys. Lett. A 177 (1993) 345–350. T.E. Simos and G. Tougelidis, A Numerov-type method for computing eigenvalues and resonances of the radial Schr¨odinger equation, Comput. Chem. 20 (1996) 397. R.M. Thomas, Phase properties of high order, almost P-stable formulae, BIT 24 (1984) 225–238. R.M. Thomas, T.E. Simos and G.V. Mitsou, A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schr¨odinger equation, University of Manchester/UMIST Joint Numerical Analysis Report No. 249 (1994). G. Vanden Berghe, V. Fack and H.E. De Meyer, Numerical methods for solving radial Schr¨odinger equation, J. Comput. Appl. Math. 29 (1989) 391–401.