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Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation. (English) Zbl 1063.65067
Summary: Multiderivative methods with minimal phase-lag are introduced, for the numerical solution of the one-dimensional Schrödinger equation. The methods are called multiderivative since they use derivatives of order two, four or six. Numerical application of the newly introduced method to the resonance problem of the one-dimensional Schrödinger equation shows its efficiency compared with other similar well-known methods of the literature.

MSC:
65L10Boundary value problems for ODE (numerical methods)
34B05Linear boundary value problems for ODE
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References:
[1] Avdelas, G.; Kefalidis, E.; Simos, T. E.: New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation. J. math. Chem. 31, 371-404 (2002) · Zbl 1078.65061
[2] Avdelas, G.; Konguetsof, A.; Simos, T. E.: A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 1. Development of the basic method. J. math. Chem. 29, 281-291 (2001) · Zbl 1022.81008
[3] Avdelas, G.; Konguetsof, A.; Simos, T. E.: A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 2. Development of the generator. Optimized generator and numerical results, J. Math. chem. 29, 293-305 (2001) · Zbl 1022.81009
[4] Avdelas, G.; Simos, T. E.: Embedded eighth order methods for the numerical solution of the Schrödinger equation. J. math. Chem. 26, 327-341 (1999) · Zbl 0954.65061
[5] Blatt, J. M.: Practical points concerning the solution of the Schrödinger equation. J. comput. Phys. 1, 382-396 (1967) · Zbl 0182.49702
[6] Chawla, M. M.: Numerov made explicit has better stability. Bit 24, 117-118 (1984) · Zbl 0568.65042
[7] Chawla, M. M.; Rao, P. S.: A numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems, vol. II, explicit method. J. comput. Appl. math. 15, 329-337 (1986) · Zbl 0598.65054
[8] Coleman, J. P.: Numerical methods for y″=$f(x,y)$ via rational approximation for the cosine. IMA J. Numer. anal. 9, 145-165 (1989) · Zbl 0675.65072
[9] Coleman, J. P.: Numerical methods for y″=$f(x,y)$. Proceedings of the first international colloquim on numerical analysis, 27-38 (1992)
[10] Cooley, J. W.: An improved eigenvalue corrector formula for solving Schrödinger’s equation for central fields. Math. comput. 15, 363-374 (1961) · Zbl 0122.35902
[11] Dormand, J. R.; El-Mikkawy, M. E.; Prince, P. J.: Families of Runge -- Kutta -- Nyström formulae. IMA J. Numer. anal. 7, 423-430 (1987) · Zbl 0627.65085
[12] Dormand, J. R.; El-Mikkawy, M. E. A.; Prince, P. J.: High-order embedded Runge -- Kutta -- Nyström formulae. IMA J. Numer. anal. 7, 595-617 (1987) · Zbl 0627.65085
[13] Herzberg, G.: Spectra of diatomic molecules. (1950)
[14] Ixaru, L. Gr.: Numerical methods for differential equations and applications. (1984) · Zbl 0543.65047
[15] Ixaru, L. Gr.; Micu, M.: Topics in theoretical physics. (1978)
[16] Ixaru, L. Gr.; Rizea, M.: A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. phys. Comm. 19, 23-27 (1980)
[17] Kalogiratou, Z.; Simos, T. E.: Construction of trigonometrically and exponentially fitted Runge -- Kutta -- Nyström methods for the numerical solution of the Schrödinger equation and related problems. J. math. Chem. 31, 211-232 (2002) · Zbl 1002.65077
[18] Konguetsof, A.; Simos, T. E.: On the construction of exponentially-fitted methods for the numerical solution of the Schrödinger equation. J. comput. Meth. sci. Eng. 1, 143-165 (2001) · Zbl 1012.65075
[19] Lambert, J. D.; Watson, I. A.: Symmetric multistep methods for periodic initial values problems. J. inst. Math. appl. 18, 189-202 (1976) · Zbl 0359.65060
[20] Landau, L. D.; Lifshitz, F. M.: Quantum mechanics. (1965) · Zbl 0178.57901
[21] I. Prigogine, S. Rice (Eds.), Advances in Chemical Physics, Vol. 93: New Methods in Computational Quantum Mechanics, Wiley, New York, 1997.
[22] Raptis, A. D.: Exponential multistep methods for ordinary differential equations. Bull. Greek math. Soc. 25, 113-126 (1984) · Zbl 0592.65046
[23] Raptis, A. D.; Allison, A. C.: Exponential-Fitting methods for the numerical solution of the Schrödinger equation. Comput. phys. Comm. 14, 1-5 (1978)
[24] T.E. Simos, Numerical solution of ordinary differential equations with periodical solution, Doctoral Dissertation, National Technical University of Athens, Greece, 1990 (in Greek).
[25] Simos, T. E.: Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem. J. math. Chem. 21, 359-372 (1997) · Zbl 0900.81031
[26] Simos, T. E.: Some embedded modified Runge -- Kutta methods for the numerical solution of some specific Schrödinger equations. J. math. Chem. 24, 23-37 (1998) · Zbl 0915.65084
[27] Simos, T. E.: A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation. J. math. Chem. 25, 65-84 (1999) · Zbl 0954.65064
[28] Simos, T. E.: A.hinchliffeatomic structure computations in chemical modelling: applications and theory, UMIST, the royal society of chemistry. Atomic structure computations in chemical modelling: applications and theory, UMIST, the royal society of chemistry, 38-142 (2000)
[29] Simos, T. E.: A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation. J. math. Chem. 27, 343-356 (2000) · Zbl 1014.81012
[30] Simos, T. E.: Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems. Chemical modellingapplication and theory, vol. 2, the royal society of chemistry 2, 170-270 (2002)
[31] Simos, T. E.: A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems. J. math. Chem. 34, 39-58 (2003) · Zbl 1029.65079
[32] Simos, T. E.: Multiderivative methods for the numerical solution of the Schrödinger equation. MATCH comm. Math. comput. Chem. 45, 7-26 (2004) · Zbl 1056.65073
[33] Simos, T. E.; Vigo-Aguiar, J.: A modified phase-fitted Runge -- Kutta method for the numerical solution of the Schrödinger equation. J. math. Chem. 30, 121-131 (2001) · Zbl 1003.65082
[34] Simos, T. E.; Vigo-Aguiar, J.: Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation. J. math. Chem. 31, 135-144 (2002) · Zbl 1002.65078
[35] Simos, T. E.; Williams, P. S.: On finite difference methods for the solution of the Schrödinger equation. Comput. chem. 23, 513-554 (1999) · Zbl 0940.65082
[36] Simos, T. E.; Williams, P. S.: A new Runge -- Kutta -- Nyström method with phase-lag of order infinity for the numerical solution of the Schrödinger equation. MATCH comm. Math. comput. Chem. 45, 123-137 (2002) · Zbl 1026.65054
[37] Tselios, K.; Simos, T. E.: Symplectic methods for the numerical solution of the radial shrodinger equation. J. math. Chem. 34, 83-94 (2003) · Zbl 1029.65134
[38] Vigo-Aguiar, J.; Simos, T. E.: A family of P-stable eighth algebraic order methods with exponential Fitting facilities. J. math. Chem. 29, 177-189 (2001) · Zbl 0995.81506
[39] Vigo-Aguiar, J.; Simos, T. E.: Family of twelve steps exponentially Fitting symmetric multistep methods for the numerical solution of the Schrödinger equation. J. math. Chem. 32, 257-270 (2002) · Zbl 1014.81055