A new implicit six-step P-stable method for the numerical solution of Schrödinger equation

References

• A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation, J. Comput. Phys. 6 (1970), pp. 378–391.
   Google Scholar
• I. Alolyan and T.E. Simos, High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation, J. Math. Chem. 48(4) (2010), pp. 925–958.
   Web of Science ®Google Scholar
• I. Alolyan and T.E. Simos, Mulitstep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation, J. Math. Chem. 48(4) (2010), pp. 1092–1143.
   Web of Science ®Google Scholar
• I. Alolyan and T.E. Simos, A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation, Comput. Math. Appl. 62(10) (2011), pp. 3756–3774.
   Web of Science ®Google Scholar
• I. Alolyan and T.E. Simos, An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems, J. Math. Chem. 54(4) (2016), pp. 1010–1040.
   Web of Science ®Google Scholar
• R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams, J. Chem. Phys. 33 (1960), pp. 795–804.
   Web of Science ®Google Scholar
• R.B. Bernstein, A. Dalgarno, H. Massey, and I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules, Proc. R. Soc. Ser. A 274 (1963), pp. 427–442.
   Google Scholar
• 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(2) (1986), pp. 233–236.
   Web of Science ®Google Scholar
• J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl., Math. 6(1) (1980), pp. 19–26.
   Google Scholar
• F. Hui and T.E. Simos, A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation, J. Math. Chem. 53(10) (2015), pp. 2191–2213.
   Web of Science ®Google Scholar
• F. Hui and T.E. Simos, Runge-Kutta type tenth algebraic order method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation, Appl. Math. Inf. Sci., 10(1) (2016), pp. 143–153.
   Google Scholar
• L.G. Ixaru and M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, J. Comput. Phys. Commun. 19(1) (1980), pp. 23–27.
   Web of Science ®Google Scholar
• J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976), pp. 189–202.
   Google Scholar
• K. Mu and T.E. Simos, A Runge-Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation, J. Math. Chem. 53(5) (2015), pp. 1239–1256.
   Web of Science ®Google Scholar
• G.D. Quinlan and S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, Astron. J. 100 (1990), pp. 1694–1700.
   Web of Science ®Google Scholar
• H. Ramos and J. Vigo-Aguiar, Variable stepsize Störmer-Cowell methods, Math. Comput. Model. 42(7–8) (2005), pp. 837–846.
   Google Scholar
• H. Ramos and J. Vigo-Aguiar, A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comput. Appl. Math. 204(1) (2007), pp. 124–136.
   Web of Science ®Google Scholar
• H. Ramos and J. Vigo-Aguiar, On the frequency choice in trigonometrically fitted methods, Appl. Math. Lett. 23(11) (2010), pp. 1378–1381.
   Web of Science ®Google Scholar
• A.D. Raptis, Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control, J. Comput. Phys. Commun. 28(4) (1983), pp. 427–431.
   Web of Science ®Google Scholar
• A.D. Raptis and J.R. Cash, A variable step method for the numerical integration of the one-dimensional Schrödinger equation, Comput. Phys. Commun. 36(2) (1985), pp. 113–119.
   Web of Science ®Google Scholar
• A. Shokri, A new eight-order symmetric two-step multiderivative method for the numerical solution of second-order IVPs with oscillating solutions, Numer. Algorithms 77(1) (2018), pp. 95–109.
   Web of Science ®Google Scholar
• A. Shokri and H. Saadat, Trigonometrically fitted high-order predictor-corrector method with phase-lag of order infinity for the numerical solution of radial Schrödinger equation, J. Math. Chem. 52(7) (2014), pp. 1870–1894.
   Web of Science ®Google Scholar
• 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(2) (2015), pp. 337–354.
   Web of Science ®Google Scholar
• A. Shokri, J. Vigo-Aguiar, M. Mehdizadeh Khalsaraei, and R. Garcia-Rubio, A new four-step P-stable Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation, J. Comput. Appl. Math. doi:10.1016/j.cam.2018.04.024.
   Web of Science ®Google Scholar
• T.E. Simos, Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems, Comput. Mater. Sci. 18(3–4) (2000), pp. 315–332.
   Web of Science ®Google Scholar
• T.E. Simos and P.S. Williams, A finite difference method for the numerical solution of the Schrödinger equation, J. Comput. Appl. Math. 79(2) (1997), pp. 189–205.
   Web of Science ®Google Scholar
• J. Vigo-Aguiar and H. Ramos, A family of A-stable Runge-Kutta collocation methods of higher order for initial-value problems, IMA J. Numer. Anal. 27(4) (2007), pp. 798–817.
   Web of Science ®Google Scholar
• J. Vigo-Aguiar and H. Ramos, A numerical ODE solver that preserves the fixed points and their stability, J. Comput. Appl. Math. 235(7) (2011), pp. 1856–1867.
   Web of Science ®Google Scholar
• J. Vigo-Aguiar and H. Ramos, On the choice of the frequency in trigonometrically-fitted methods for periodic problems, J. Comput. Appl. Math. 277 (2015), pp. 94–105.
   Web of Science ®Google Scholar
• Z. Wang, P-stable linear symmetric multistep methods for periodic initial-value problems, Comput. Phys. Commun. 171(3) (2005), pp. 162–174.
   Web of Science ®Google Scholar
• B. Wang, Triangular splitting implementation of RKN-type Fourier collocation methods for second-order differential equations, Math. Methods Appl. Sci. 41(5) (2018), pp. 1998–2011.
   Web of Science ®Google Scholar
• B. Wang, A. Iserles, and X. Wu, Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems, Found. Comput. Math. 16(1) (2016), pp. 151–181.
   Web of Science ®Google Scholar
• B. Wang, F. Meng, and Y. Fang, Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations, Appl. Numer. Math. 119 (2017), pp. 164–178.
   Web of Science ®Google Scholar
• B. Wang, X. Wu, and F. Meng, Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations, J. Comput. Appl. Math. 313 (2017), pp. 185–201.
   Web of Science ®Google Scholar
• B. Wang, X. Wu, F. Meng, and Y. Fang, Exponential Fourier collocation methods for solving first-order differential equations, J. Comput. Math. 35(6) (2017), pp. 711–736.
   Web of Science ®Google Scholar
• B. Wang, H. Yang, and F. Meng, Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations, Calcolo 54(1) (2017), pp. 117–140.
   Web of Science ®Google Scholar
• X. Xi and T.E. Simos, A new four-stages twelfth algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation, MATCH Commun. Math. Comput. Chem. 77(2) (2017), pp. 333–392.
   Web of Science ®Google Scholar
• X. You, Y. Zhou, and X. Cheng, A novel family of P-stable symmetric extended linear multistep methods for oscillators, Appl. Math. Comput. 249 (2014), pp. 597–610.
   Web of Science ®Google Scholar
• Z. Zhou and T.E. Simos, A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation, J. Math. Chem. 54(2) (2016), pp. 442–465.
   Web of Science ®Google Scholar