Construction of variablestepsize multistep formulas
Author:
Robert D. Skeel
Journal:
Math. Comp. 47 (1986), 503510, S45
MSC:
Primary 65L05
MathSciNet review:
856699
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Abstract: A systematic way of extending a general fixedstepsize multistep formula to a minimum storage variablestepsize formula has been discovered that encompasses fixedcoefficient (interpolatory), variablecoefficient (variable step), and fixed leading coefficient as special cases. In particular, it is shown that the "interpolatory" stepsize changing technique of Nordsieck leads to a truly variablestepsize multistep formula (which has implications for local error estimation and formula changing), and it is shown that the "variablestep" stepsize changing technique applicable to the Adams and backwarddifferentiation formulas has a reasonable generalization to the general multistep formula. In fact, it is shown how to construct a variableorder family of variablecoefficient formulas. Finally, it is observed that the first Dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration.
 [1]
G.
D. Byrne and A.
C. Hindmarsh, A polyalgorithm for the numerical solution of
ordinary differential equations, ACM Trans. Math. Software
1 (1975), no. 1, 71–96. MR 0378432
(51 #14600)
 [2]
Germund
Dahlquist, Convergence and stability in the numerical integration
of ordinary differential equations, Math. Scand. 4
(1956), 33–53. MR 0080998
(18,338d)
 [3]
G. G. Dahlquist, On Stability and Error Analysis for Stiff Nonlinear Problems, Part 1, Report TRITANA7508, Dept. of Computer Sci., Royal Inst. of Technology, Stockholm, 1975.
 [4]
Ȧke
Björck and Germund
Dahlquist, Numerical methods, PrenticeHall, Inc., Englewood
Cliffs, N.J., 1974. Translated from the Swedish by Ned Anderson;
PrenticeHall Series in Automatic Computation. MR 0368379
(51 #4620)
 [5]
Germund
G. Dahlquist, Werner
Liniger, and Olavi
Nevanlinna, Stability of twostep methods for variable integration
steps, SIAM J. Numer. Anal. 20 (1983), no. 5,
1071–1085. MR 714701
(85b:65079), http://dx.doi.org/10.1137/0720076
 [6]
J. Descloux, A Note on a Paper by A. Nordsieck, Report #131, Dept. of Computer Sci., Univ. of Illinois, UrbanaChampaign, 1963.
 [7]
C. Dill & C. W. Gear, "A graphical search for stiffly stable methods for ordinary differential equations," J. Assoc. Comput. Mach., v. 18, 1971, pp. 7579.
 [8]
C.
William Gear, Numerical initial value problems in ordinary
differential equations, PrenticeHall, Inc., Englewood Cliffs, N.J.,
1971. MR
0315898 (47 #4447)
 [9]
C.
W. Gear and D.
S. Watanabe, Stability and convergence of variable order multistep
methods, SIAM J. Numer. Anal. 11 (1974),
1044–1058. MR 0368437
(51 #4678)
 [10]
A. Hindmarsh, Documentation for LSODE, Math. & Stats. Section L300, Lawrence Livermore Laboratory, Livermore, Calif., 1980.
 [11]
K.
R. Jackson and R.
SacksDavis, An alternative implementation of variable stepsize
multistep formulas for stiff ODEs, ACM Trans. Math. Software
6 (1980), no. 3, 295–318. MR 585340
(81m:65120), http://dx.doi.org/10.1145/355900.355903
 [12]
M. D. Kregel & J. M. Heimerl, Comments on the Solution of Coupled Stiff Differential Equations, Proc. of the 1977 Army Numerical Analysis and Computers Conference, Report No. 773, U. S. Army Research Office, Research Triangle Park, N. C., 1977, pp. 553563.
 [13]
Fred
T. Krogh, Algorithms for changing the step size, SIAM J.
Numer. Anal. 10 (1973), 949–965. MR 0356515
(50 #8985)
 [14]
Arnold
Nordsieck, On numerical integration of ordinary
differential equations, Math. Comp. 16 (1962), 22–49. MR 0136519
(24 #B2552), http://dx.doi.org/10.1090/S00255718196201365195
 [15]
J. Sand, Stability and Boundedness Results for VariableStep VariableFormula Methods, Report TRITANA8219, Dept. of Numer. Anal. and Computer Sci., Royal Inst. of Technology, Stockholm, 1982.
 [16]
L. F. Shampine, How to Live with a Reasonable ODE Code (DIFSUB) until a Good One Arrives, manuscript, ca. 1974.
 [17]
L.
F. Shampine and M.
K. Gordon, Computer solution of ordinary differential
equations, W. H. Freeman and Co., San Francisco, Calif., 1975. The
initial value problem. MR 0478627
(57 #18104)
 [18]
R. D. Skeel, Convergence of Multivalue Methods for Solving Ordinary Differential Equations, Report TR7316, Dept. of Computing Sci., Univ. of Alberta, Edmonton, 1973.
 [19]
Robert
D. Skeel, Equivalent forms of multistep
formulas, Math. Comp. 33
(1979), no. 148, 1229–1250.
MR 537967
(80j:65027), http://dx.doi.org/10.1090/S00255718197905379674
 [20]
R. D. Skeel & T. V. Vu, "Note on blended linear multistep methods." (submitted.)
 [21]
C.
S. Wallace and G.
K. Gupta, General linear multistep methods to solve ordinary
differential equations, Austral. Comput. J. 5 (1973),
62–69. MR
0362919 (50 #15357)
 [1]
 G. D. Byrne & A. C. Hindmarsh, "A polyalgorithm for the numerical solution of ordinary differential equations," ACM Trans. Math. Software, v. 1, 1975, pp. 7196. MR 0378432 (51:14600)
 [2]
 G. G. Dahlquist, "Numerical integration of ordinary differential equations," Math. Scand., v. 4, 1956, pp. 3350. MR 0080998 (18:338d)
 [3]
 G. G. Dahlquist, On Stability and Error Analysis for Stiff Nonlinear Problems, Part 1, Report TRITANA7508, Dept. of Computer Sci., Royal Inst. of Technology, Stockholm, 1975.
 [4]
 G. G. Dahlquist & A. Björck, Numerical Methods (transl. by N. Anderson), PrenticeHall, Englewood Cliffs, N. J., 1975. MR 0368379 (51:4620)
 [5]
 G. G. Dahlquist, W. Liniger & O. Nevanlinna, "Stability of twostep methods for variable integration steps," SIAM J. Numer. Anal., v. 20, 1983, pp. 10711085. MR 714701 (85b:65079)
 [6]
 J. Descloux, A Note on a Paper by A. Nordsieck, Report #131, Dept. of Computer Sci., Univ. of Illinois, UrbanaChampaign, 1963.
 [7]
 C. Dill & C. W. Gear, "A graphical search for stiffly stable methods for ordinary differential equations," J. Assoc. Comput. Mach., v. 18, 1971, pp. 7579.
 [8]
 C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0315898 (47:4447)
 [9]
 C. W. Gear & D. S. Watanabe, "Stability and convergence of variable order multistep methods," SIAM J. Numer. Anal., v. 11, 1974, pp. 10441058. MR 0368437 (51:4678)
 [10]
 A. Hindmarsh, Documentation for LSODE, Math. & Stats. Section L300, Lawrence Livermore Laboratory, Livermore, Calif., 1980.
 [11]
 K. R. Jackson & R. SacksDavis, "An alternative implementation of variable stepsize multistep formulas for stiff ODEs," ACM Trans. Math. Software, v. 6, 1980, pp. 295318. MR 585340 (81m:65120)
 [12]
 M. D. Kregel & J. M. Heimerl, Comments on the Solution of Coupled Stiff Differential Equations, Proc. of the 1977 Army Numerical Analysis and Computers Conference, Report No. 773, U. S. Army Research Office, Research Triangle Park, N. C., 1977, pp. 553563.
 [13]
 F. T. Krogh, "Algorithms for changing the step size," SIAM J. Numer. Anal., v. 10, 1973, pp. 949965. MR 0356515 (50:8985)
 [14]
 A. Nordsieck, "On numerical integration of ordinary differential equations," Math. Comp., v. 16, 1962, pp. 2249. MR 0136519 (24:B2552)
 [15]
 J. Sand, Stability and Boundedness Results for VariableStep VariableFormula Methods, Report TRITANA8219, Dept. of Numer. Anal. and Computer Sci., Royal Inst. of Technology, Stockholm, 1982.
 [16]
 L. F. Shampine, How to Live with a Reasonable ODE Code (DIFSUB) until a Good One Arrives, manuscript, ca. 1974.
 [17]
 L. F. Shampine & M. K. Gordon, Computer Solution of Ordinary Differential Equations, Freeman, San Francisco, 1975. MR 0478627 (57:18104)
 [18]
 R. D. Skeel, Convergence of Multivalue Methods for Solving Ordinary Differential Equations, Report TR7316, Dept. of Computing Sci., Univ. of Alberta, Edmonton, 1973.
 [19]
 R. D. Skeel, "Equivalent forms of multistep formulas," Math. Comp., v. 33, 1979, pp. 12291250; Corrigendum, ibid., v. 47, 1986, p. 769. MR 537967 (80j:65027)
 [20]
 R. D. Skeel & T. V. Vu, "Note on blended linear multistep methods." (submitted.)
 [21]
 C. S. Wallace & G. K. Gupta, "General linear multistep methods to solve ordinary differential equations," Austral. Comput. J., v. 5, 1973, pp. 6269. MR 0362919 (50:15357)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819860856699X
PII:
S 00255718(1986)0856699X
Keywords:
Multistep formula,
multistep method,
multivalue method,
variable stepsize,
variable order
Article copyright:
© Copyright 1986
American Mathematical Society
