Correction in the dominant space: a numerical technique for a certain class of stiff initial value problems

Authors:
P. Alfeld and J. D. Lambert

Journal:
Math. Comp. **31** (1977), 922-938

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1977-0519719-2

Correction:
Math. Comp. **31** (1977), 922-938.

Original Article:
Math. Comp. **31** (1977), 922-938.

MathSciNet review:
0519719

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider a stiff linear initial value problem , where the eigenvalues of may be separated into two sets, one of which dominates the other. The dominant eigenvalues and corresponding right and left eigenvectors may be computed by the power method. A technique is proposed which consists of taking one forward step by a conventional multistep method and then making a correction entirely in the subspace spanned by the eigenvectors corresponding to the dominant eigenvalues. A number of alternative corrections are proposed and discussed. It is shown that the technique is stable provided that the product of the steplength and each of the subdominant eigenvalues lies within the region of absolute stability of the multistep method. The application of the technique to nonlinear problems is discussed, and numerical results are reported.

**[1]**M. CLINT & A. JENNINGS, "A simultaneous iteration method for the unsymmetric eigenvalue problem",*J. Inst. Math. Appl.*, v. 8, 1971, pp. 111-121. MR**45**#6174. MR**0297116 (45:6174)****[2]**C. W. CRYER, "A new class of highly-stable methods: -stable methods,"*BIT*, v. 13, 1973, pp. 153-159. MR**48**#1469. MR**0323111 (48:1469)****[3]**G. G. DAHLQUIST, "A special stability problem for linear multistep methods,"*BIT*, v. 3, 1963, pp. 27-43. MR**30**#715. MR**0170477 (30:715)****[4]**G. G. DAHLQUIST, "A numerical method for some ordinary differential equations with large Lipschitz constants,"*Information Processing*68. Vol. 1 (Proc. IFIP Congress, Edinburgh, 1968), A. J. H. Morell, Editor, North-Holland, Amsterdam, 1969, pp. 183-186. MR**41**#2937. MR**0258290 (41:2937)****[5]**G. G. DAHLQUIST, "Problems related to the numerical treatment of stiff differential systems,"*ACM Proc. International Computing Symposium*, North-Holland, Amsterdam, 1974.**[6]**W. GAUTSCHI, "Numerical integration of ordinary differential equations based on trigonometric polynomials,"*Numer. Math.*, v. 3, 1961, pp. 381-397. MR**25**#1647. MR**0138200 (25:1647)****[7]**C. W. GEAR,*Numerical Initial Value Problems in Ordinary Differential Equations*, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR**47**#4447. MR**0315898 (47:4447)****[8]**J. D. LAMBERT, "The numerical integration of a special class of stiff differential systems,"*Proc. Fifth Manitoba Conf. on Numerical Mathematics*, Univ. of Manitoba, Oct. 1975. MR**0408253 (53:12018)****[9]**L. ODEN,*An Experimental and Theoretical Analysis of the SAPS Method for Stiff Ordinary Differential Equations*, Report NA 71.28, Dept. of Inform. Proc., Royal Inst. of Tech., Stockholm, 1971.**[10]**G. W. STEWART,*Simultaneous Iteration for Computing Invariant Subspaces of Non-Hermitian Matrices*, Gatlinburg VI; Springer-Verlag, 1974. MR**0400677 (53:4508)****[11]**O. B. WIDLUND, "A note on unconditionally stable linear multistep methods,"*BIT*, v. 7, 1967, pp. 65-70. MR**35**#6373. MR**0215533 (35:6373)**

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DOI:
https://doi.org/10.1090/S0025-5718-1977-0519719-2

Article copyright:
© Copyright 1977
American Mathematical Society