Error growth analysis via stability regions for discretizations of initial value problems

BIT Numerical Mathematics

Volumn 37, Issue 2, 1997, Pages 442-464

  Spijker, M N ^a   Straetemans, F A J ^a  

^a LEIDEN UNIVERSITY   (Netherlands)

Author keywords

Discretization; Error growth; Initial value problem; Numerical method; Resolvent condition; Stability analysis; Stability region

Indexed keywords

EID: 0347833395     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02510222     Document Type: Article

References (23)

1. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, John Wiley, Chicester, 1987.
2. M. Crouzeix, S. Larsson, S. Piskarev, and V. Thomée, The stability of rational approximations of analytic semigroups, BIT, 33 (1993), pp. 74-84.
3. J. L. M. van Dorsselaer, J. F. B. M. Kraaijevanger, and M. N. Spijker, Linear stability analysis in the numerical solution of initial value problems, Acta Numerica, 1993, pp. 199-237.
4. M. B. Giles, Stability and convergence of discretizations of initial value p.d.e.'s, Report 96/06, Oxford University Comp. Lab., 1996.
5. D. F. Griffiths, I. Christie, and A. R. Mitchell, Analysis of error growth for explicit difference schemes in conduction-convection problems, Internat. J. Numer. Meth. Engrg, 15 (1980), pp. 1075-1081.
6. Ch. Grossmann and H.-G. Roos, Numerik partieller Differentialgleichungen, Teubner, Stuttgart, 1992.
7. E. Hairer and G. Wanner, Solving Ordinary Differential Equations, Vol. II, Springer, Berlin, 1991.
8. P. J. van der Houwen, Construction of Integration Formulas for Initial Value Problems, North-Holland, Amsterdam, New York, Oxford, 1977.
9. H. W. J. Lenferink and M. N. Spijker, A generalization of the numerical range of a matrix, Linear Algebra Appl., 140 (1990), pp. 251-266.
10. H. W. J. Lenferink and M. N. Spijker, On a generalization of the resolvent condition in the Kreiss matrix theorem, Math. Comp., 57 (1991), pp. 211-220.
11. H. W. J. Lenferink and M. N. Spijker, On the use of stability regions in the numerical analysis of initial value problems, Math. Comp., 57 (1991), pp. 221-237.
12. C. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), pp. 293-313.
13. K. W. Morton, Stability of finite-difference approximations to a diffusion-convection equation, Int. J. Num. Meth. Engrg., 15 (1980), pp. 677-683.
14. J. Noye, Computational Techniques for Differential Equations, North-Holland, Amsterdam, 1984.
15. C. Palencia, Stability of rational multistep approximations of holomorphic semi-groups, Math. Comp., 64 (1995), pp. 591-599.
16. S. V. Parter, Stability, convergence, and pseudo-stability of finite-difference equations for an over-determined problem, Numer. Math., 4 (1962), pp. 277-292.
17. S. C. Reddy and L. N. Trefethen, Stability of the method of lines, Numer. Math., 62 (1992), pp. 235-267.
18. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., John Wiley and Sons, New York, 1967.
19. M. N. Spijker, Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems, Math. Comp., 45 (1985), pp. 377-392.
20. M. N. Spijker, Numerical ranges and stability estimates, Appl. Numer. Math. 13 (1993), pp. 241-249.
21. M. N. Spijker and F. A. J. Straetemans, Stability estimates for families of matrices of nonuniformly bounded order, Linear Algebra Appl., 239 (1996), pp. 77-102.
22. M. N. Spijker and F. A. J. Straetemans, A note on the order of contact between sets in the complex plane, Report TW-96-06, Mathem. Instit., Leiden University, 1996.
23. G. Stoyan, On Monotone Difference Schemes for Weakly Coupled Systems of Partial Differential Equations, Computational Mathematics, Banach Center Publications, Vol. 13, PWN-Polish Scientific Publishers, Warsaw, 1984.