Numerical stability, resolvent conditions and delay differential equations

Applied Numerical Mathematics

Volumn 24, Issue 2-3, 1997, Pages 233-246

  Spijker, M N ^a  

^a LEIDEN UNIVERSITY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; ESTIMATION; PROBLEM SOLVING;

DELAY DIFFERENTIAL EQUATIONS; ERROR GROWTH ANALYSIS; KREISS RESOLVENT CONDITION; LINEAR INITIAL VALUE PROBLEMS;

DIFFERENTIAL EQUATIONS;

EID: 0031210214     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(97)00023-8     Document Type: Article

References (22)

1. V.K. Barwell, Special stability problems for functional differential equations, BIT 15 (1975) 130-135.
2. M. Calvo and T. Grande, On the asymptotic stability of θ-methods for delay differential equations, Numer. Math. 54 (1988) 257-269.
3. G. Dahlquist, H. Mingyou and R. LeVeque, On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods, Numer. Math. 42 (1983) 1-13.
4. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II (Springer, Berlin, 1991).
5. R.A. Horn and C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1990).
6. J.F.B.M. Kraaijevanger, Two counterexamples related to the Kreiss matrix theorem, BIT 34 (1994) 113-119.
7. H.O. Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2 (1962) 153-181.
8. R.J. LeVeque and L.N. Trefethen, On the resolvent condition in the Kreiss matrix theorem, BIT 24 (1984) 584-591.
9. M.Z. Liu and M.N. Spijker, The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. Numer. Anal. 10 (1990) 31-48.
10. C. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991) 293-313.
11. K.W. Morton, On a matrix theorem due to H.O. Kreiss, Comm. Pure Appl. Math. 17 (1964) 375-379.
12. C. Palencia, Stability of rational multistep approximations of holomorphic semigroups, Math. Comp. 64 (1995) 591-599.
13. S.V. Parter, Stability, convergence, and pseudo-stability of finite-difference equations for an overdetermined problem, Numer. Math. 4 (1962) 277-292.
14. R.D. Richtmyer and K.W. Morton, Difference Methods for Initial Value Problems (Wiley, New York, 2nd ed., 1967).
15. M.N. Spijker, On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem, BIT 31 (1991) 551-555.
16. M.N. Spijker and F.A.J. Straetemans, Stability estimates for families of matrices of nonuniformly bounded order, Linear Algebra Appl. 239 (1996) 77-102.
17. 2-stability, the resolvent condition, and strict H-stability, Linear Algebra Appl. 41 (1981) 151-159.
18. E. Tadmor, The resolvent condition and uniform powerboundedness, Linear Algebra Appl. 80 (1986) 250-252.
19. 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) 199-237.
20. D.S. Watanabe and M.G. Roth, The stability of difference formulas for delay differential equations, SIAM J. Numer. Anal. 22 (1985) 132-145.
21. E. Wegert and L.N. Trefethen, From the Buffon needle problem to the Kreiss matrix theorem, Amer. Math. Monthly 101 (1994) 132-139.
22. M. Zennaro, P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math. 49 (1986) 305-318.