Global errors of numerical ODE solvers and Lyapunov’s theory of stability

IMA Journal of Numerical Analysis

Volumn 21, Issue 1, 2001, Pages 387-406

  Viswanath, Divakar ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DYNAMICAL SYSTEMS; ERROR ANALYSIS; INITIAL VALUE PROBLEMS; STABILITY;

EXACT SOLUTION; GLOBAL ERRORS; NONLINEAR PROBLEMS; NUMERICAL ODE SOLVERS; QUASI-PERIODIC; STABILITY PROPERTIES; THEORETICAL BOUNDS; TIME STEP;

NUMERICAL METHODS;

EID: 0035529771     PISSN: 02724979     EISSN: None     Source Type: Journal    
DOI: 10.1093/imanum/21.1.387     Document Type: Article

References (27)

1. BELLMAN, R. 1953 Stability Theory of Differential Equations. New York: McGraw-Hill.
2. CALVO, M. P. & HAIRER, E. 1995 Accurate long-term integration of dynamical systems. Appl. Numer.Math. 18, 95-105.
3. CALVO, M. P. & SANZ-SERNA, J. M. The development of variable-step symplectic integrators, with application to the two-body problem. SIAM J. Sci. Comput. 14.
4. CANO, B. & SANZ-SERNA, J. M. 1997 Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems. SIAM J. Numer. Anal. 34, 1391-1417.
5. ESTEP, D. J. & STUART, A. M. 1995 The rate of error growth in Hamiltonian-conserving integrators. Z. Agnew. Math. Phys. 46, 407-418.
6. GRAGG, W. B. 1964 Repeated extrapolation to the limit in the numerical solution of ordinary differential equations, Thesis, University of California; see also 1965 SIAM J. Numer. Anal. 2, 384-403.
7. GRAGG, W. B. 1964 Repeated extrapolation to the limit in the numerical solution of ordinary differential equations, Thesis, University of California; see also 1965 SIAM J. Numer. Anal. 2, 384-403.
8. HAIRER, E., NØRSETT, S. P., & WANNER, G. 1980 Solving Ordinary Differential Equations I. New York: Springer.
9. HALE, J. 1969 Ordinary Differential Equations. New York: John Wiley.
10. HARTMAN, P. 1973 Ordinary Differential Equations. New York: John Wiley.
11. HENRICI, P. 1962 Discrete Variable Methods in Ordinary Differential Equations. New York: John Wiley.
12. HENRICI, P. 1963 Error Propagation for Difference Methods. New York: John Wiley.
13. HIRSH, M. W., PUGH, C. C., & SHUB, M. 1977 Invariant Manifolds. Lecture notes in Mathematics 583. New York: Springer.
14. ISERLES, A. & SÖDERLIND, G. 1993 Global bounds on numerical error for ordinary differential equations. J. Complexity 9, 97-112.
15. LYAPUNOV, A. M. 1892 Problème générale de la stabilité du mouvement. Comm. Soc. Math. Kharkov 2; 1893 Ibid 3; 1907 Ann. Fac. Sci. Toulouse 9, 204-474; 1949 Ann. Math. Studies, Vol. 17, Princeton: Princeton University Press.
16. LYAPUNOV, A. M. 1892 Problème générale de la stabilité du mouvement. Comm. Soc. Math. Kharkov 2; 1893 Ibid 3; 1907 Ann. Fac. Sci. Toulouse 9, 204-474; 1949 Ann. Math. Studies, Vol. 17, Princeton: Princeton University Press.
17. LYAPUNOV, A. M. 1892 Problème générale de la stabilité du mouvement. Comm. Soc. Math. Kharkov 2; 1893 Ibid 3; 1907 Ann. Fac. Sci. Toulouse 9, 204-474; 1949 Ann. Math. Studies, Vol. 17, Princeton: Princeton University Press.
18. LYAPUNOV, A. M. 1892 Problème générale de la stabilité du mouvement. Comm. Soc. Math. Kharkov 2; 1893 Ibid 3; 1907 Ann. Fac. Sci. Toulouse 9, 204-474; 1949 Ann. Math. Studies, Vol. 17, Princeton: Princeton University Press.
19. MALKIN, J. G. 1956 Theory of stability of motion. Atomic Energy Commision Technical Report, 3352.
20. QUISPEL, G. R. W. & DYT, C. P. 1998 Volume-preserving integrators have linear error growth. Phys. Lett. A 242, 25-30.
21. ROBINSON, C. 1995 Dynamical Systems. Stability, Symbolic Dynamics, and Chaos. Boca Roton: CRC press.
22. RUDIN, W. 1987 Real and Complex Analysis, 3rd edn, New York: McGraw-Hill.
23. STETTER, H. J. 1973 Analysis of Discretization Methods for Ordinary Differential Equations. New York: Springer.
24. STOFFER, D. & NIPP, K. 1991 Invariant curves for variable step size integrators. BIT 31, 169-180.
25. STUART, A. M. 1997 Probabilistic and deterministic convergence proofs for software for initial value problems. Numerical Algorithms 14, 227-260.
26. STUART, A. M. & HUMPHRIES, A. R. 1996 Dynamical Systems and Numerical Analysis. Cambridge: Cambridge University Press.
27. YOSHIZAWA, T. 1966 Stability Theory by Liapunov's Second Method. Tokyo: Mathematical Society of Japan.