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About stability estimates and resolvent conditions

Authors 1, 1, 1

Affiliations

  1. Department of Mathematics and Computer Science, University of Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

Bibliography

  1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York 1965.
  2. F. F. Bonsall and J. Duncan, Numerical ranges, in: Studies in Functional Analysis, R. G. Bartle (ed.), Mathematical Association of America, 1980, 1-49.
  3. Ph. Brenner and V. Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), 683-694.
  4. J. B. Conway, A Course in Functional Analysis, Springer, New York 1985.
  5. M. Crouzeix, On multistep approximation of semigroups in Banach spaces, J. Comput. Appl. Math. 20 (1987), 25-35.
  6. D. F. Griffiths, I. Christie and A. R. Mitchell, Analysis of error growth for explicit difference schemes in conduction-convection problems, Internat. J. Numer. Methods Engrg. 15 (1980), 1075-1081.
  7. R. D. Grigorieff, Time discretization of semigroups by the variable two-step BDF method, in: Numerical Treatment of Differential Equations, K. Strehmel (ed.), Teubner, Leipzig 1991, 204-216.
  8. P. Henrici, Applied and Computational Complex Analysis, vol. 2, Wiley, New York 1977.
  9. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, 1990.
  10. J. F. B. M. Kraaijevanger, H. W. J. Lenferink and M. N. Spijker, Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations, J. Comput. Appl. Math. 20 (1987), 67-81.
  11. H. O. Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2 (1962), 153-181.
  12. H. O. Kreiss, Well posed hyperbolic initial boundary value problems and stable difference approximations, in: Proc. Third Internat. Conf. on Hyperbolic Problems, Uppsala, Sweden, 1990.
  13. H. O. Kreiss and L. Wu, On the stability definition of difference approximations for the initial boundary value problem, Appl. Numer. Math. 12 (1993), 213-227.
  14. G. I. Laptev, Conditions for the uniform well-posedness of the Cauchy problem for systems of equations, Soviet Math. Dokl. 16 (1975), 65-69.
  15. H. W. J. Lenferink and M. N. Spijker, The relevance of stability regions in the numerical solution of initial value problems, in: Numerical Treatment of Differential Equations, K. Strehmel (ed.), Teubner, Leipzig 1988, 95-103.
  16. H. W. J. Lenferink and M. N. Spijker, A generalization of the numerical range of a matrix, Linear Algebra Appl. 140 (1990), 251-266.
  17. 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), 221-237.
  18. H. W. J. Lenferink and M. N. Spijker, On a generalization of the resolvent condition in the Kreiss matrix theorem, ibid., 211-220.
  19. R. J. LeVeque and L. N. Trefethen, On the resolvent condition in the Kreiss matrix theorem, BIT 24 (1984), 584-591.
  20. Ch. Lubich, On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), 839-853.
  21. Ch. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), 293-313.
  22. C. A. McCarthy and J. Schwartz, On the norm of a finite boolean algebra of projections, and applications to theorems of Kreiss and Morton, Comm. Pure Appl. Math. 18 (1965), 191-201.
  23. J. Miller, On the resolvent of a linear operator associated with a well-posed Cauchy problem, Math. Comp. 22 (1968), 541-548.
  24. J. Miller and G. Strang, Matrix theorems for partial differential and difference equations, Math. Scand. 18 (1966), 113-123.
  25. K. W. Morton, On a matrix theorem due to H. O. Kreiss, Comm. Pure Appl. Math. 17 (1964), 375-379.
  26. O. Nevanlinna, Remarks on time discretization of contraction semigroups, Helsinki Univ. Techn., Inst. Math., report HTKK-MAT-A225 (1984).
  27. S. V. Parter, Stability, convergence, and pseudo-stability of finite-difference equations for an over-determined problem, Numer. Math. 4 (1962), 277-292.
  28. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York 1983.
  29. C. Pearcy, An elementary proof of the power inequality for the numerical radius, Michigan Math. J. 13 (1966), 289-291.
  30. S. C. Reddy and L. N. Trefethen, Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues, Comput. Methods Appl. Mech. Engrg. 80 (1990), 147-164.
  31. S. C. Reddy and L. N. Trefethen, Stability of the method of lines, Numer. Math. 62 (1992), 235-267.
  32. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed., Wiley, New York 1967.
  33. M.-N. Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), 919-931.
  34. W. Rudin, Functional Analysis, McGraw-Hill, New York 1973.
  35. J. C. Smith, An inequality for rational functions, Amer. Math. Monthly 92 (1985), 740-741.
  36. M. N. Spijker, Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems, Math. Comp. 45 (1985), 377-392.
  37. M. N. Spijker, On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem, BIT 31 (1991), 551-555.
  38. E. Tadmor, The equivalence of L2-stability, the resolvent condition and strict H-stability, Linear Algebra Appl. 41 (1981), 151-159.
  39. L. N. Trefethen, Non-normal matrices and pseudospectra, in preparation.
  40. E. Wegert and L. N. Trefethen, From the Buffon needle problem to the Kreiss matrix theorem, Amer. Math. Monthly, to appear.
Banach Center Publications
1994/29/1
Export to PBN, Opens in new tab
Pages:
215-225
Main language of publication
English
Published
1994
Exact and natural sciences