A class of stable advanced multistep methods
Authors:
Jack Williams and Frank de Hoog
Journal:
Math. Comp. 28 (1974), 163177
MSC:
Primary 65L05
MathSciNet review:
0356519
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Abstract: A class of Astable advanced multistep methods is derived for the numerical solution of initial value problems in ordinary differential equations. The methods, of all orders of accuracy up to ten, only require values of y' and are self starting. Results for the asymptotic behaviour of the discretization error and for estimating local truncation error are also obtained. The practical implementation of the fourth order method is described and the method applied to some stiff equations. Numerical comparisons are made with Gear's method.
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F. H. Chipman, "Astable RungeKutta processes," Nordisk Tidskr. Informationsbehandling (BIT), v. 11, 1971, pp. 384388. MR 0295582 (45:4648)
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G. Dahlquist, "A special stability problem for linear multistep methods," Nordisk Tidskr. Informationsbehandling, v. 3, 1963, pp. 2743. MR 30 #715. MR 0170477 (30:715)
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G. Dahlquist et al., Survey of Stiff Ordinary Differential Equations, The Royal Institute of Technology, Stockholm, Report NA 70.11, 1970.
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J. W. Daniel, NonLinear Equations Arising in Deferred Correction of Initial Value Problems, MRC Technical Report No. 818, 1967; Also in Acta Ci. Venezolana, v. 19, 1968, pp. 123128. (Spanish) MR 40 #8269. MR 0255062 (40:8269)
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J. W. Daniel, V. Pereyra & L. L. Schumaker, Iterated Deferred Corrections for Initial Value Problems, MRC Technical Report #808, Madison, Wis., 1967; Also in Acta Ci. Venezolana, v. 19, 1968, pp. 128135. (Spanish) MR 40 #8270. MR 0255063 (40:8270)
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B. L. Ehle, "High order Astable methods for the numerical solution of systems of D.E.'s," Nordisk Tidskr. Informationsbehandling (BIT), v. 8, 1968, pp. 276278. MR 39 #1119. MR 0239762 (39:1119)
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C. W. Gear, The Automatic Integration of Stiff Ordinary Differential Equations, Proc. IFIP Congress, Supplement, Booklet A: 8185, 1968. MR 0260180 (41:4808)
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C. W. Gear, "DIFSUB for solution of ordinary differential equations," Comm. ACM, v. 14, 1971, pp. 185190.
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C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0315898 (47:4447)
 [11]
P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 24 #B1772. MR 0135729 (24:B1772)
 [12]
B. L. Hulme, "Discrete Galerkin and related onestep methods for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 881891. MR 0315899 (47:4448)
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F. T. Krogh, On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations, Jet Propulsion Laboratory, Pasadena, Calif., Tech. Mem. No. 217, 1970.
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L. Lapidus & J. H. Seinfeld, Numerical Solution of Ordinary Differential Equations, Math. in Sci. and Engineering, vol. 74, Academic Press, New York and London, 1971. MR 43 #7073. MR 0281355 (43:7073)
 [15]
A. Ralston, A First Course in Numerical Analysis, McGrawHill, New York, 1965. MR 32 #8479. MR 0191070 (32:8479)
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J. B. Rosser, "A RungeKutta for all seasons," SIAM Rev., v. 9, 1967, pp. 417452. MR 36 #2325. MR 0219242 (36:2325)
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L. F. Shampine & H. A. Watts, "Block implicit onestep methods," Math. Comp., v. 23, 1969, pp. 731740. MR 41 #9445. MR 0264854 (41:9445)
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H. A. Watts & L. F. Shampine, "Astable block implicit onestep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 12, 1972, pp. 252266. MR 0307483 (46:6603)
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H. A. Watts, AStable Block Implicit OneStep Methods, Sandia Laboratories Report SCRR71 0296.
 [20]
K. Wright, "Some relationships between implicit RungeKutta, collocation and Lanczos methods, and their stability properties," Nordisk Tidskr. Informationsbehandling (BIT), v. 10, 1970, pp. 217227. MR 42 #1345. MR 0266439 (42:1345)
 [1]
 O. Axelsson, "A class of Astable methods." Nordisk Tidskr. Informationsbehandling (BIT), v. 9, 1969, pp. 185199. MR 40 #8266. MR 0255059 (40:8266)
 [2]
 F. H. Chipman, "Astable RungeKutta processes," Nordisk Tidskr. Informationsbehandling (BIT), v. 11, 1971, pp. 384388. MR 0295582 (45:4648)
 [3]
 G. Dahlquist, "A special stability problem for linear multistep methods," Nordisk Tidskr. Informationsbehandling, v. 3, 1963, pp. 2743. MR 30 #715. MR 0170477 (30:715)
 [4]
 G. Dahlquist et al., Survey of Stiff Ordinary Differential Equations, The Royal Institute of Technology, Stockholm, Report NA 70.11, 1970.
 [5]
 J. W. Daniel, NonLinear Equations Arising in Deferred Correction of Initial Value Problems, MRC Technical Report No. 818, 1967; Also in Acta Ci. Venezolana, v. 19, 1968, pp. 123128. (Spanish) MR 40 #8269. MR 0255062 (40:8269)
 [6]
 J. W. Daniel, V. Pereyra & L. L. Schumaker, Iterated Deferred Corrections for Initial Value Problems, MRC Technical Report #808, Madison, Wis., 1967; Also in Acta Ci. Venezolana, v. 19, 1968, pp. 128135. (Spanish) MR 40 #8270. MR 0255063 (40:8270)
 [7]
 B. L. Ehle, "High order Astable methods for the numerical solution of systems of D.E.'s," Nordisk Tidskr. Informationsbehandling (BIT), v. 8, 1968, pp. 276278. MR 39 #1119. MR 0239762 (39:1119)
 [8]
 C. W. Gear, The Automatic Integration of Stiff Ordinary Differential Equations, Proc. IFIP Congress, Supplement, Booklet A: 8185, 1968. MR 0260180 (41:4808)
 [9]
 C. W. Gear, "DIFSUB for solution of ordinary differential equations," Comm. ACM, v. 14, 1971, pp. 185190.
 [10]
 C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0315898 (47:4447)
 [11]
 P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 24 #B1772. MR 0135729 (24:B1772)
 [12]
 B. L. Hulme, "Discrete Galerkin and related onestep methods for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 881891. MR 0315899 (47:4448)
 [13]
 F. T. Krogh, On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations, Jet Propulsion Laboratory, Pasadena, Calif., Tech. Mem. No. 217, 1970.
 [14]
 L. Lapidus & J. H. Seinfeld, Numerical Solution of Ordinary Differential Equations, Math. in Sci. and Engineering, vol. 74, Academic Press, New York and London, 1971. MR 43 #7073. MR 0281355 (43:7073)
 [15]
 A. Ralston, A First Course in Numerical Analysis, McGrawHill, New York, 1965. MR 32 #8479. MR 0191070 (32:8479)
 [16]
 J. B. Rosser, "A RungeKutta for all seasons," SIAM Rev., v. 9, 1967, pp. 417452. MR 36 #2325. MR 0219242 (36:2325)
 [17]
 L. F. Shampine & H. A. Watts, "Block implicit onestep methods," Math. Comp., v. 23, 1969, pp. 731740. MR 41 #9445. MR 0264854 (41:9445)
 [18]
 H. A. Watts & L. F. Shampine, "Astable block implicit onestep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 12, 1972, pp. 252266. MR 0307483 (46:6603)
 [19]
 H. A. Watts, AStable Block Implicit OneStep Methods, Sandia Laboratories Report SCRR71 0296.
 [20]
 K. Wright, "Some relationships between implicit RungeKutta, collocation and Lanczos methods, and their stability properties," Nordisk Tidskr. Informationsbehandling (BIT), v. 10, 1970, pp. 217227. MR 42 #1345. MR 0266439 (42:1345)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403565198
PII:
S 00255718(1974)03565198
Keywords:
Astable,
advanced multistep methods,
stiff systems
Article copyright:
© Copyright 1974
American Mathematical Society
