stable Obrechkoff methods with minimal phaselag for periodic initial value problems
Author:
U. Anantha Krishnaiah
Journal:
Math. Comp. 49 (1987), 553559
MSC:
Primary 65L05
MathSciNet review:
906188
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Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper Pstable methods of and with minimal phaselag (frequency distortion) are derived. Numerical results for both linear and nonlinear problems are presented.
 [1]
U.
Anantha Krishnaiah, A class of twostep Pstable methods for the
accurate integration of second order periodic initial value problems,
J. Comput. Appl. Math. 14 (1986), no. 3,
455–459. MR
831087 (87e:65046), http://dx.doi.org/10.1016/03770427(86)900804
 [2]
L. Brusa & L. Nigro, "A onestep method for direct integration of structural dynamic equations," Internat. J. Numer. Methods Engrg., v. 15, 1980, pp. 685699.
 [3]
J.
R. Cash, High order 𝑃stable formulae for the numerical
integration of periodic initial value problems, Numer. Math.
37 (1981), no. 3, 355–370. MR 627110
(82j:65044), http://dx.doi.org/10.1007/BF01400315
 [4]
M.
M. Chawla, Twostep fourth order 𝑃stable methods for
second order differential equations, BIT 21 (1981),
no. 2, 190–193. MR 627879
(82h:65051), http://dx.doi.org/10.1007/BF01933163
 [5]
M.
M. Chawla and P.
S. Rao, A Noumerovtype method with minimal phaselag for the
integration of second order periodic initialvalue problems, J.
Comput. Appl. Math. 11 (1984), no. 3, 277–281.
MR 777103
(86c:65061), http://dx.doi.org/10.1016/03770427(84)900025
 [6]
Germund
Dahlquist, On accuracy and unconditional stability of linear
multistep methods for second order differential equations, BIT
18 (1978), no. 2, 133–136. MR 499228
(80a:65145), http://dx.doi.org/10.1007/BF01931689
 [7]
Walter
Gautschi, Numerical integration of ordinary differential equations
based on trigonometric polynomials, Numer. Math. 3
(1961), 381–397. MR 0138200
(25 #1647)
 [8]
I.
Gladwell and R.
M. Thomas, Damping and phase analysis for some methods for solving
secondorder ordinary differential equations, Internat. J. Numer.
Methods Engrg. 19 (1983), no. 4, 495–503. MR 702055
(84g:65088), http://dx.doi.org/10.1002/nme.1620190404
 [9]
E.
Hairer, Unconditionally stable methods for second order
differential equations, Numer. Math. 32 (1979),
no. 4, 373–379. MR 542200
(80f:65082), http://dx.doi.org/10.1007/BF01401041
 [10]
M.
K. Jain, R.
K. Jain, and U.
Anantha Krishnaiah, 𝑃stable methods for periodic initial
value problems of second order differential equations, BIT
19 (1979), no. 3, 347–355. MR 548614
(80h:65048), http://dx.doi.org/10.1007/BF01930988
 [11]
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. MR 0431691
(55 #4686)
 [12]
E.
Stiefel and D.
G. Bettis, Stabilization of Cowell’s method, Numer.
Math. 13 (1969), 154–175. MR 0263250
(41 #7855)
 [13]
R.
M. Thomas, Phase properties of high order, almost 𝑃stable
formulae, BIT 24 (1984), no. 2, 225–238.
MR 753550
(86a:65066), http://dx.doi.org/10.1007/BF01937488
 [14]
R. van Dooren, "Stabilization of Cowell's classical finite difference method for numerical integration," J. Comput. Phys., v. 16, 1974, pp. 186192.
 [1]
 U. Ananthakrishnaiah, "A class of twostep Pstable methods for the accurate integration of second order periodic initial value problems," J. Comput. Appl. Math., v. 14, 1986, pp. 455459. MR 831087 (87e:65046)
 [2]
 L. Brusa & L. Nigro, "A onestep method for direct integration of structural dynamic equations," Internat. J. Numer. Methods Engrg., v. 15, 1980, pp. 685699.
 [3]
 J. R. Cash, "High order Pstable formulae for the numerical integration of periodic initial value problems," Numer. Math., v. 37, 1981, pp. 355370. MR 627110 (82j:65044)
 [4]
 M. M. Chawla, "Twostep fourth order Pstable methods for second order differential equations," BIT, v. 21, 1981, pp. 190193. MR 627879 (82h:65051)
 [5]
 M. M. Chawla & P. S. Rao, "A Numerovtype method with minimal phaselag for the integration of second order periodic initial value problems," J. Comput. Appl. Math., v. 11, 1984, pp. 277281. MR 777103 (86c:65061)
 [6]
 G. Dahlquist, "On accuracy and unconditional stability of linear multistep methods for second order differential equations," BIT, v. 18, 1978, pp. 133136. MR 499228 (80a:65145)
 [7]
 W. Gautschi, "Numerical integration of ordinary differential equations based on trigonometric polynomials," Numer. Math., v. 3, 1961, pp. 381397. MR 0138200 (25:1647)
 [8]
 I. Gladwell & R. M. Thomas, "Damping and phase analysis for some methods for solving second order ordinary differential equations," Internat. J. Numer. Methods Engrg., v. 19, 1983, pp. 493503. MR 702055 (84g:65088)
 [9]
 E. Hairer, "Unconditionally stable methods for second order differential equations," Numer. Math., v. 32, 1979, pp. 373379. MR 542200 (80f:65082)
 [10]
 M. K. Jain, R. K. Jain & U. Ananthakrishnaiah, "Pstable methods for periodic initial value problems of second order differential equations," BIT, v. 19, 1979, pp. 347355. MR 548614 (80h:65048)
 [11]
 J. D. Lambert & I. A. Watson, "Symmetric multistep methods for periodic initial value problems," J. Inst. Math. Appl., v. 18, 1976, pp. 189202. MR 0431691 (55:4686)
 [12]
 E. Stiefel & D. G. Bettis, "Stabilization of Cowell's methods," Numer. Math., v. 13, 1969, pp. 154175. MR 0263250 (41:7855)
 [13]
 R. M. Thomas, "Phase properties of high order almost Pstable formulae," BIT, v. 24, 1984, pp. 225238. MR 753550 (86a:65066)
 [14]
 R. van Dooren, "Stabilization of Cowell's classical finite difference method for numerical integration," J. Comput. Phys., v. 16, 1974, pp. 186192.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819870906188X
PII:
S 00255718(1987)0906188X
Keywords:
Pstable,
Obrechkoff,
phaselag,
periodic initial value problems,
secondorder differential equations,
undamped Duffing's equation
Article copyright:
© Copyright 1987
American Mathematical Society
