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A four-step phase-fitted method for the numerical integration of second order initial-value problems. (English) Zbl 0726.65089

A four-step method with phase-lag of infinite order is developed for the numerical integration of second order initial-value problems of the form: \(y''(x)=f(x,y),\quad y(x_ 0)=y_ 0,\quad y'(x_ 0)=y'_ 0.\) Examples occur in celestial mechanics, in quantum mechanical scattering problems and elsewhere.
The idea is to maintain a free parameter \(\alpha\) in the method such that the method to be fitted to an oscillatory component of the theoretical solution. Applications of the new method have been done in two problems.
The first is the “almost periodic” problem studied by E. Stiefel and D. G. Bettis [Numer. Math. 13, 154-175 (1969; Zbl 0219.65062)]: \(z''+z=0.001e^{ix},\quad z(0)=1,\quad z'(0)=0.9995i,\quad z\in C\) and the other is the resonance problem of the one-dimensional Schrödinger equation: \(y''(x)=f(x)y(x),\) \(x\in [0,\infty)\), with \(f(x)=W(x)-E,\) \(W(x)=\ell (\ell +1)/x^ 2+V(x),\ell \in {\mathbb{Z}}\), E is the energy (E\(\in {\mathbb{R}})\). In both problems the new suggested method is more accurate than other methods with minimal phase-lag, especially for large step-sizes.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C25 Periodic solutions to ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Citations:

Zbl 0219.65062
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Full Text: DOI

References:

[1] M. M. Chawla and P. S. Rao,A Numerov-type method with minimal phase-lag for the numerical integration of second order periodic initial-value problem, J. Comput. Appl. Math. 11 (1984), 277–281. · Zbl 0565.65041 · doi:10.1016/0377-0427(84)90002-5
[2] M. M. Chawla and P. S. Rao,A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problem. II. Explicit method, J. Comput. Appl. Math. 15 (1986), 329–337. · Zbl 0598.65054
[3] 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), 233–236. · Zbl 0596.65047 · doi:10.1016/0377-0427(86)90094-4
[4] M. M. Chawla and P. S. Rao,An explicit sixth-order method with phase-lag or order eight for y”=f(t,y), J. Comput. Appl. Math. 17 (1987), 365–368. · Zbl 0614.65084 · doi:10.1016/0377-0427(87)90113-0
[5] R. M. Thomas,Phase properties of high order, almost P-stable formulae, BIT 24 (1984), 225–238. · Zbl 0569.65052 · doi:10.1007/BF01937488
[6] P. J. Van der Houwen and B. P. Sommeijer,Predictor-corrector methods for periodic second-order initial-value problems, IMA J. Num. Anal. 7 (1987), 407–422. · Zbl 0631.65074 · doi:10.1093/imanum/7.4.407
[7] J. P. Coleman,Numerical methods for y”=f(x,y) via rational approximations for the cosine, IMA J. Num. Anal. 9 (1989), 145–165. · Zbl 0675.65072 · doi:10.1093/imanum/9.2.145
[8] E. Stiefel and D. G. Bettis,Stabilization of Cowell’s method, Num. Math. 13 (1969), 154–175. · Zbl 0219.65062 · doi:10.1007/BF02163234
[9] P. Henrici,Discrete Variable Methods in Ordinary Differential Equations, (1962), 311, John Wiley and Sons. · Zbl 0112.34901
[10] R. K. Jain, N. S. Kambo and R. Goel,A sixth-order P-stable symmetric multistep method for periodic initial-value problems of second-order differential equations, IMA J. Num. Anal. 4 (1984), 117–125. · Zbl 0539.65053 · doi:10.1093/imanum/4.1.117
[11] A. D. Raptis,Two-step methods for the numerical solution of the Schrödinger equation, Computing 28 (1982), 373–378. · Zbl 0473.65060 · doi:10.1007/BF02279820
[12] R. A. Buckingham,Numerical Solution of Ordinary and Partial Differential Equations, (1962), Pergamon Press. · Zbl 0129.45904
[13] J. W. Cooley,An improved eigenvalue corrector formula for solving the Schrödinger equation for central fields, Math. Comput. 15 (1961), 363–374. · Zbl 0122.35902
[14] L. Brusa and L. Nigro,A one-step method for direct integration of structural dynamic equations, Internat. J. Numer. Meth. Engng. 15 (1980), 685–699. · Zbl 0426.65034 · doi:10.1002/nme.1620150506
[15] W. Liniger and R. A. Willoughby,Efficient integration methods for stiff systems of ordinary differential equations, SIAM J. Num. Anal. 7 (1970), 47–66. · Zbl 0187.11003 · doi:10.1137/0707002
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