Block implicit onestep methods
Author:
Daniel S. Watanabe
Journal:
Math. Comp. 32 (1978), 405414
MSC:
Primary 65L05
MathSciNet review:
0494959
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A new class of block implicit onestep methods for ordinary differential equations is presented. The methods are based on quadrature and generate function values at nonmesh points through Hermite interpolation. A general convergence theorem for block implicit methods is given, and the stability of the new class of methods is analyzed. The class contains Astable, stiffly stable, strongly Astable, and strongly stiffly stable methods. Numerical results demonstrating the efficiency and effectiveness of a particular block method are presented.
 [1]
George
D. Andria, George
D. Byrne, and David
R. Hill, Natural spline block implicit methods, Nordisk
Tidskr. Informationsbehandling (BIT) 13 (1973),
131–144. MR 0323110
(48 #1468)
 [2]
D. BARTON, I. M. WILLERS & R. V. M. ZAHAR, "Taylor series methods for ordinary differential equationsan evaluation," in Mathematical Software (J. R. Rice, Editor), Academic Press, New York, 1971, pp. 369390.
 [3]
C.
G. Broyden, A new method of solving nonlinear simultaneous
equations, Comput. J. 12 (1969/1970), 94–99. MR 0245197
(39 #6509)
 [4]
J.
C. Butcher, Implicit RungeKutta
processes, Math. Comp. 18 (1964), 50–64. MR 0159424
(28 #2641), http://dx.doi.org/10.1090/S00255718196401594249
 [5]
F. H. CHIPMAN, Numerical Solution of Initial Value Problems Using AStable RungeKutta Processes, Ph. D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1971.
 [6]
B. L. EHLE, On Padé Approximations to the Exponential Function and AStable Methods for the Numerical Solution of Initial Value Problems, Ph. D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1969.
 [7]
C.
William Gear, Numerical initial value problems in ordinary
differential equations, PrenticeHall Inc., Englewood Cliffs, N.J.,
1971. MR
0315898 (47 #4447)
 [8]
C. HERMITE, "Sur la formule d'interpolation de Lagrange," J. Reine Angew. Math., v. 84, 1878, pp. 7079.
 [9]
Bernie
L. Hulme, Discrete Galerkin and related onestep
methods for ordinary differential equations, Math. Comp. 26 (1972), 881–891. MR 0315899
(47 #4448), http://dx.doi.org/10.1090/S00255718197203158998
 [10]
J.
Barkley Rosser, A RungeKutta for all seasons, SIAM Rev.
9 (1967), 417–452. MR 0219242
(36 #2325)
 [11]
L.
F. Shampine and H.
A. Watts, Block implicit onestep
methods, Math. Comp. 23 (1969), 731–740. MR 0264854
(41 #9445), http://dx.doi.org/10.1090/S00255718196902648545
 [12]
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)
 [13]
H.
A. Watts and L.
F. Shampine, 𝐴stable block implicit onestep methods,
Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972),
252–266. MR 0307483
(46 #6603)
 [14]
Jack
Williams and Frank
de Hoog, A class of 𝐴stable advanced
multistep methods, Math. Comp. 28 (1974), 163–177. MR 0356519
(50 #8989), http://dx.doi.org/10.1090/S00255718197403565198
 [1]
 G. D. ANDRIA, G. D. BYRNE & D. R. HILL, "Natural spline block implicit methods," BIT, v. 13, 1973, pp. 131144. MR 0323110 (48:1468)
 [2]
 D. BARTON, I. M. WILLERS & R. V. M. ZAHAR, "Taylor series methods for ordinary differential equationsan evaluation," in Mathematical Software (J. R. Rice, Editor), Academic Press, New York, 1971, pp. 369390.
 [3]
 C. G. BROYDEN, "A new method of solving nonlinear simultaneous equations," Comput. J., v. 12, 1969, pp. 9499. MR 0245197 (39:6509)
 [4]
 I. C. BUTCHER, "Implicit RungeKutta processes," Math. Comp., v. 18, 1964, pp. 5064. MR 0159424 (28:2641)
 [5]
 F. H. CHIPMAN, Numerical Solution of Initial Value Problems Using AStable RungeKutta Processes, Ph. D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1971.
 [6]
 B. L. EHLE, On Padé Approximations to the Exponential Function and AStable Methods for the Numerical Solution of Initial Value Problems, Ph. D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1969.
 [7]
 C. W. GEAR, Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0315898 (47:4447)
 [8]
 C. HERMITE, "Sur la formule d'interpolation de Lagrange," J. Reine Angew. Math., v. 84, 1878, pp. 7079.
 [9]
 B. L. HULME, "Discrete Galerkin and related onestep methods for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 881891. MR 0315899 (47:4448)
 [10]
 J. B. ROSSER, "A RungeKutta for all seasons," SIAM Rev., v. 9, 1967, pp. 417452. MR 0219242 (36:2325)
 [11]
 L. F. SHAMPINE & H. A. WATTS, "Block implicit onestep methods," Math. Comp., v. 23, 1969, pp. 731740. MR 0264854 (41:9445)
 [12]
 H. J. STETTER, "Economical global error estimation," in Stiff Differential Systems (R. A. Willoughby, Editor), Plenum Press, New York, 1974, pp. 245258. MR 0405863 (53:9655)
 [13]
 H. A. WATTS & L. F. SHAMPINE, "Astable block implicit onestep methods," BIT, v. 12, 1972, pp. 252266. MR 0307483 (46:6603)
 [14]
 J. WILLIAMS & F. de HOOG, "A class of Astable advanced multistep methods," Math. Comp., v. 28, 1974, pp. 163177. MR 0356519 (50:8989)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65L05
Retrieve articles in all journals
with MSC:
65L05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804949590
PII:
S 00255718(1978)04949590
Keywords:
Ordinary differential equations,
methods based on quadrature,
Hermite interpolation,
Astable,
stiffly stable,
strongly Astable,
strongly stiffly stable
Article copyright:
© Copyright 1978 American Mathematical Society
