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

Full-text PDF

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]**Maurice Clint and Alan Jennings,*A simultaneous iteration method for the unsymmetric eigenvalue problem*, J. Inst. Math. Appl.**8**(1971), 111–121. MR**0297116****[2]**Colin W. Cryer,*A new class of highly-stable methods: 𝐴₀-stable methods*, Nordisk Tidskr. Informationsbehandling (BIT)**13**(1973), 153–159. MR**0323111****[3]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[4]**Germund Dahlquist,*A numerical method for some ordinary differential equations with large Lipschitz constants*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 183–186. MR**0258290****[5]**G. G. DAHLQUIST, "Problems related to the numerical treatment of stiff differential systems,"*ACM Proc. International Computing Symposium*, North-Holland, Amsterdam, 1974.**[6]**Walter Gautschi,*Numerical integration of ordinary differential equations based on trigonometric polynomials*, Numer. Math.**3**(1961), 381–397. MR**0138200**, https://doi.org/10.1007/BF01386037**[7]**C. William Gear,*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898****[8]**J. D. Lambert,*The numerical integration of a special class of stiff differential systems*, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 91–108. Congressus Numerantium, No. XVI. MR**0408253****[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*, Numer. Math.**25**(1975/76), no. 2, 123–136. MR**0400677**, https://doi.org/10.1007/BF01462265**[11]**Olof B. Widlund,*A note on unconditionally stable linear multistep methods*, Nordisk Tidskr. Informations-Behandling**7**(1967), 65–70. MR**0215533**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L05

Retrieve articles in all journals with MSC: 65L05

Additional Information

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

Article copyright:
© Copyright 1977
American Mathematical Society