An extension of the applicability of iterated deffered corrections
Authors:
Reinhard Frank, Joerg Hertling and Christoph W. Ueberhuber
Journal:
Math. Comp. 31 (1977), 907915
MSC:
Primary 65L10
MathSciNet review:
0445848
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Abstract: A new way of estimating local discretization errors (based on an idea due to P. E. Zadunaisky) is introduced. If error estimates obtained by this method are used in connection with the general class of iterated deferred correction algorithms, they lead to an extension of the domain of applicability, when compared with the variants used by Fox and Pereyra.
 [1]
J. W. DANIEL & A. J. MARTIN, Implementing Deferred Corrections for Numerov's Difference Method for SecondOrder TwoPoint BoundaryValue Problems, Report CNA107, Center for Numerical Analysis, Univ. of Texas, Austin, Texas, 1975.
 [2]
L.
Fox, The numerical solution of twopoint boundary problems in
ordinary differential equations, Oxford University Press, New York,
1957. MR
0102178 (21 #972)
 [3]
Numerical solution of ordinary and partial differential
equations., Based on a Summer School held in Oxford, AugustSeptember
1961, Pergamon Press, OxfordLondonParis; AddisonWesley Publishing Co.,
Inc., Reading, Mass.Palo Alto, Calif.London, 1962. MR 0146969
(26 #4488)
 [4]
Reinhard
Frank, The method of iterated defectcorrection and its application
to twopoint boundary value problems. I, Numer. Math.
25 (1975/76), no. 4, 409–419. MR 0445846
(56 #4180a)
 [5]
R. FRANK, J. HERTLING & C. W. UEBERHUBER, Iterated Defect Correction Based on Estimates of the Local Discretization Error, Report No. 18/76, Inst. für Numer. Math., Tech. Univ., Vienna, 1976.
 [6]
R. FRANK & C. W. UEBERHUBER, Iterated Defect Correction for RungeKutta Methods, Report No. 14/75, Inst. für Numer. Math., Tech. Univ., Vienna, 1975.
 [7]
Herbert
B. Keller, Numerical solution of two point boundary value
problems, Society for Industrial and Applied Mathematics,
Philadelphia, Pa., 1976. Regional Conference Series in Applied Mathematics,
No. 24. MR
0433897 (55 #6868)
 [8]
M.
Lentini and V.
Pereyra, Boundary problem solvers for first order systems based on
deferred corrections, Numerical solutions of boundary value problems
for ordinary differential equations (Proc. Sympos., Univ. Maryland,
Baltimore, Md., 1974), Academic Press, New York, 1975,
pp. 293–315. MR 0488787
(58 #8297)
 [9]
B. LINDBERG, Error Estimation and Iterative Improvement for the Numerical Solution of Operator Equations, Report UIUCDCSR76820, Dept. of Comput. Sci., Univ. of Illinois at UrbanaChampaign, Urbana, Illinois, July, 1976.
 [10]
Victor
Pereyra, On improving an approximate solution of a functional
equation by deferred corrections, Numer. Math. 8
(1966), 376–391. MR 0203967
(34 #3814)
 [11]
Victor
Pereyra, Iterated deferred corrections for nonlinear operator
equations, Numer. Math. 10 (1967), 316–323. MR 0221760
(36 #4812)
 [12]
Victor
Pereyra, Iterated deferred corrections for nonlinear boundary value
problems, Numer. Math. 11 (1968), 111–125. MR 0225498
(37 #1091)
 [13]
V. L. PEREYRA, High Order Finite Difference Solution of Differential Equations, Report STANCS73348, Comput. Sci. Dept., Stanford University, Stanford, California, 1973.
 [14]
Hans
J. Stetter, Economical global error estimation, Stiff
differential systems (Proc. Internat. Sympos., Wildbad, 1973), Plenum, New
York, 1974, pp. 245–258. IBM Res. Sympos. Ser. MR 0405863
(53 #9655)
 [15]
P. E. ZADUNAISKY, "A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations," Proc. Internat. Astronom. Union Symposium No. 25 (Thessaloniki, 1964), Academic Press, New York, 1966.
 [1]
 J. W. DANIEL & A. J. MARTIN, Implementing Deferred Corrections for Numerov's Difference Method for SecondOrder TwoPoint BoundaryValue Problems, Report CNA107, Center for Numerical Analysis, Univ. of Texas, Austin, Texas, 1975.
 [2]
 L. FOX, The Numerical Solution of TwoPoint Boundary Value Problems in Ordinary Differential Equations, Oxford Univ. Press, New York, 1957. MR 21 #972. MR 0102178 (21:972)
 [3]
 L. FOX (Editor), Numerical Solution of Ordinary and Partial Differential Equations, Pergamon Press, Oxford, 1962, pp. 3839. MR 26 #4488. MR 0146969 (26:4488)
 [4]
 R. FRANK, "The method of iterated defectcorrection and its application to twopoint boundary value problems," Part I, Numer. Math., v. 25, 1976, pp. 409419; ibid., Part II, Numer. Math., v. 27, 1977, pp. 407420. MR 0445846 (56:4180a)
 [5]
 R. FRANK, J. HERTLING & C. W. UEBERHUBER, Iterated Defect Correction Based on Estimates of the Local Discretization Error, Report No. 18/76, Inst. für Numer. Math., Tech. Univ., Vienna, 1976.
 [6]
 R. FRANK & C. W. UEBERHUBER, Iterated Defect Correction for RungeKutta Methods, Report No. 14/75, Inst. für Numer. Math., Tech. Univ., Vienna, 1975.
 [7]
 H. B. KELLER, Numerical Solution of TwoPoint BoundaryValue Problems, Conference Board of the Math. Sci. Regional Conf. Ser. in Appl. Math., no 24, SIAM, Philadelphia, Pa., 1976, p. 34. MR 0433897 (55:6868)
 [8]
 M. LENTINI & V. PEREYRA, "Boundary problem solvers for first order systems based on deferred corrections," A. K. AZIZ (Editor), Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations, Academic Press, New York, 1975. MR 0488787 (58:8297)
 [9]
 B. LINDBERG, Error Estimation and Iterative Improvement for the Numerical Solution of Operator Equations, Report UIUCDCSR76820, Dept. of Comput. Sci., Univ. of Illinois at UrbanaChampaign, Urbana, Illinois, July, 1976.
 [10]
 V. L. PEREYRA, "On improving an approximate solution of a functional equation by deferred corrections," Numer. Math., v. 8, 1966, pp. 376391. MR 34 #3814. MR 0203967 (34:3814)
 [11]
 V. L. PEREYRA, "Iterated deferred corrections for nonlinear operator equations," Numer. Math., v. 10, 1967, pp. 316323. MR 36 #4812. MR 0221760 (36:4812)
 [12]
 V. L. PEREYRA, "Iterated deferred corrections for nonlinear boundary value problems," Numer. Math., v. 11, 1968, pp. 111125. MR 37 #1091. MR 0225498 (37:1091)
 [13]
 V. L. PEREYRA, High Order Finite Difference Solution of Differential Equations, Report STANCS73348, Comput. Sci. Dept., Stanford University, Stanford, California, 1973.
 [14]
 H. J. STETTER, "Economical global error estimation," R. A. WILLOUGHBY (Editor), Stiff Differential Systems (Proc. Internat. Sympos., Wildbad, Germany, 1973), Plenum Press, New York and London, 1974, pp. 245258. MR 53 #9655. MR 0405863 (53:9655)
 [15]
 P. E. ZADUNAISKY, "A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations," Proc. Internat. Astronom. Union Symposium No. 25 (Thessaloniki, 1964), Academic Press, New York, 1966.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197704458488
PII:
S 00255718(1977)04458488
Article copyright:
© Copyright 1977
American Mathematical Society
