Abstract
When solving a stiff differential system by an implicit method, factorizing the Jacobian and solving the resulting linear equations often dominate the cost. We develop some methods related to a block Schur factorization of the Jacobian for separating the stiff components. These methods use block versions of the OR or LR algorithm or, for sparse Jacobians, orthogonal iteration to derive an approximate Jacobian. The technique is practical only for systems with relatively few stiff components.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alefeld, P. and Lambert, J.D. "Correction in the dominant space: a numerical technique for a certain class of stiff initial value problems", Math. Comput. 31, 922–938 (1977).
Björck, Å. "A block OR algorithm for partitioning stiff differential systems", BIT 23:3 (1983).
Buurema, H.J. "A geometric proof of convergence for the OR method", Thesis, Rijksuniversiteit te Groningen (1970).
Carver, M.B. and MacEwen, S.R. "On the use of sparse matrix approximation to the Jacobian in integrating large sets of ordinary differential equations", SIAM J. Sci. Stat. Comput., 2, 51–64 (1981).
Curtis, A.R. "Jacobian matrix properties and their impact on choice of software for stiff ode systems", Report CSS 134, Computer Science and Systems Division, A.E.R.E Harwell, Oxfordshire (1983).
Dahlquist, G. and Söderlind, G. "Some problems related to stiff nonlinear differential systems", Computing Methods in Applied Sciences and Engineering V (eds. R. Glowinski and J.L. Lions), North-Holland Publ. Co., 57–74 (1982).
Duff, I.S. "MA28-A set of Fortran subroutines for sparse nonsymmetric linear equations. AERE Report R.8730, Harwell (1979).
Enright, W.H. and Kamel, M.S. "Automatic partitioning of stiff systems and exploiting the resultant structure" ACM Trans. Math. Software 5, 374–385 (1979).
Golub, G.H. and Van Loan, C. "Applied Matrix Computation" in preparation.
Hindmarsh, A.C. "Stiff system problems and solutions at LLNL", Report UCRL-87406, Lawrence Livermore National Laboratory, (1982).
Parlett, B.N. and Poole, W.G. "A geometric theory for the OR, LU and power iterations", SIAM J. Numer. Anal. 10, 389–412 (1973).
Robertson, H.H. "Some factors affecting the efficiency of stiff integration routines", in Numerical Software — Needs and Availability, (ed. D.A.H. Jacobs), Academic Press, 279–301 (1978).
G.W. Stewart "Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices", Numer. Math. 25, 123–136 (1976).
Söderlind, G. "On the efficient solution of nonlinear equations in numerical methods for stiff differential systems", Report TRITA-NA-8114, The Royal Inst. of Tech., Stockholm (1981).
Watkins, D.S. and HansonSmith, R.W. "The numerical solution of separably stiff systems by precise partitioning", Report Dept. Pure and Appl. Math., Washington State Univ., Pullman (1982).
Wilkinson, J.H. "The algebraic eigenvalue problem", Clarendon Press, Oxford (1965).
Zu-fan, M. "Partitioning a stiff ordinary differential system by a scaling technique", Report TRITA-NA-8210, The Royal Inst. of Tech., Stockholm (1982).
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Björck, Å. (1984). Some methods for separating stiff components in initial value problems. In: Griffiths, D.F. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099516
Download citation
DOI: https://doi.org/10.1007/BFb0099516
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13344-5
Online ISBN: 978-3-540-38881-4
eBook Packages: Springer Book Archive