Error propagation of general linear methods for ordinary differential equations

Journal of Complexity

Volumn 23, Issue 4-6, 2007, Pages 560-580

  Butcher, J C ^a   Jackiewicz, Z ^b   Wright, W M ^c  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

^b ARIZONA STATE UNIVERSITY   (United States)

^c LA TROBE UNIVERSITY   (Australia)

Author keywords

Adaptive stepsize selection; Error propagation; General linear methods; Local error estimation for methods of adjacent orders; Nordsieck representation; Stability analysis

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; ORDINARY DIFFERENTIAL EQUATIONS;

ADAPTIVE STEPSIZE SELECTION; ADJACENT ORDERS; ERROR PROPAGATION; GENERAL LINEAR METHODS; LOCAL ERROR ESTIMATION; NORDSIECK REPRESENTATION; STABILITY ANALYSIS;

LINEAR SYSTEMS;

EID: 36249002755     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2007.01.009     Document Type: Article

References (25)

1. Butcher J.C. Diagonally-implicit multi-stage integration methods. Appl. Numer. Math. 11 (1993) 347-363
2. Butcher J.C. The Numerical Solution of Ordinary Differential Equations (2003), Wiley, New York
3. Butcher J.C., Chartier P., and Jackiewicz Z. Nordsieck representation of DIMSIMs. Numer. Algorithms 16 (1997) 209-230
4. Butcher J.C., Chartier P., and Jackiewicz Z. Experiments with a variable-order type 1 DIMSIM code. Numer. Algorithms 22 (1999) 237-261
5. Butcher J.C., and Jackiewicz Z. A new approach to error estimation for general linear methods. Numer. Math. 95 (2003) 487-502
6. Butcher J.C., and Jackiewicz Z. Construction of general linear methods with Runge-Kutta stability properties. Numer. Algorithms 36 (2004) 53-72
7. Butcher J.C., and Jackiewicz Z. Unconditionally stable general linear methods for ordinary differential equations. BIT 44 (2004) 557-570
8. J.C. Butcher, Z. Jackiewicz, Towards a code for nonstiff differential systems based on general linear methods with inherent Runge-Kutta stability, in preparation.
9. Butcher J.C., and Podhaisky H. On error estimation in general linear methods for stiff ODEs. Appl. Numer. Math. 56 (2006) 345-357
10. Butcher J.C., and Wright W.M. The construction of practical general linear methods. BIT 43 (2003) 695-721
11. Chartier P. The potential of parallel multi-value methods for the simulation of large real-life problems. CWI Quarterly 11 (1998) 7-32
12. Guglielmi N., and Zennaro M. On the asymptotic properties of a family of matrices. Linear Algebra Appl. 322 (2001) 169-192
13. Guglielmi N., and Zennaro M. On the zero-stability of variable stepsize multistep methods: the spectral radius approach. Numer. Math. 88 (2001) 445-458
14. Hairer E., Nørsett S.P., and Wanner G. Solving Ordinary Differential Equations I. Nonstiff Problems (1993), Springer, Berlin, Heidelberg, New York
15. Hull T.E., Enright W.H., Fellen B.M., and Sedgwick A.E. Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9 (1972) 603-637
16. Jackiewicz Z. Implementation of DIMSIMs for stiff differential systems. Appl. Numer. Math. 42 (2002) 251-267
17. Jackiewicz Z. Construction and implementation of general linear methods for ordinary differential equations. A review. J. Sci. Comput. 25 (2005) 29-49
18. Jackiewicz Z., Podhaisky H., and Weiner R. Construction of highly stable two-step W-methods for ordinary differential equations. J. Comput. Appl. Math. 167 (2004) 389-403
19. Kirchgraber U. Multistep methods are essentially one-step methods. Numer. Math. 48 (1986) 85-90
20. Lambert J.D. Computational Methods in Ordinary Differential Equations (1973), Wiley, Chichester, New York
21. Nordsieck A. On numerical integration of ordinary differential equations. Math. Comp. 16 (1962) 22-49
22. Shampine L.F. Numerical Solution of Ordinary Differential Equations (1994), Chapman & Hall, New York, London
23. Shampine L.F., Gladwell I., and Thompson S. Solving ODEs with MATLAB (2003), Cambridge University Press, Cambridge
24. Stoffer D. General linear methods: connection to one step methods and invariant curves. Numer. Math. 64 (1993) 395-408
25. W.M. Wright, General linear methods with inherent Runge-Kutta stability, Ph.D. Thesis, The University of Auckland, New Zealand, 2003.