On the stability of variable stepsize rational approximations of holomorphic semigroups
HTML articles powered by AMS MathViewer
- by C. Palencia PDF
- Math. Comp. 62 (1994), 93-103 Request permission
Abstract:
We consider variable stepsize time approximations of holomorphic semigroups on general Banach spaces. For strongly ${\text {A}}(\theta )$-acceptable rational functions a general stability theorem is proved, which does not impose any constraint on the ratios between stepsizes.References
- Philip Brenner and Vidar Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), no. 4, 683–694. MR 537280, DOI 10.1137/0716051
- Philip Brenner, Vidar Thomée, and Lars B. Wahlbin, Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Mathematics, Vol. 434, Springer-Verlag, Berlin-New York, 1975. MR 0461121
- M. Crouzeix, S. Larsson, S. Piskarëv, and V. Thomée, The stability of rational approximations of analytic semigroups, BIT 33 (1993), no. 1, 74–84. MR 1326004, DOI 10.1007/BF01990345
- K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR 774402 N. Dunford and L. Schwartz, Linear operators, part I, Interscience, New York, 1958.
- Hector O. Fattorini, The Cauchy problem, Encyclopedia of Mathematics and its Applications, vol. 18, Addison-Wesley Publishing Co., Reading, Mass., 1983. With a foreword by Felix E. Browder. MR 692768
- Reuben Hersh and Tosio Kato, High-accuracy stable difference schemes for well-posed initial value problems, SIAM J. Numer. Anal. 16 (1979), no. 4, 670–682. MR 537279, DOI 10.1137/0716050 S. Larsson, private communication.
- Stig Larsson, Vidar Thomée, and Lars B. Wahlbin, Finite-element methods for a strongly damped wave equation, IMA J. Numer. Anal. 11 (1991), no. 1, 115–142. MR 1089551, DOI 10.1093/imanum/11.1.115 M. N. LeRoux, Semidiscretizations in time for parabolic problems, Math. Comp. 33 (1979), 919-931.
- Christian Lubich and Olavi Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), no. 2, 293–313. MR 1112225, DOI 10.1007/BF01931289
- Robert McKelvey, Spectral measures, generalized resolvents, and functions of positive type, J. Math. Anal. Appl. 11 (1965), 447–477. MR 208369, DOI 10.1016/0022-247X(65)90097-1
- C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal. 30 (1993), no. 5, 1373–1384. MR 1239826, DOI 10.1137/0730071
- C. Palencia and J. M. Sanz-Serna, An extension of the Lax-Richtmyer theory, Numer. Math. 44 (1984), no. 2, 279–283. MR 753959, DOI 10.1007/BF01410111
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
- J. M. Sanz-Serna, Stability and convergence in numerical analysis. I. Linear problems—a simple, comprehensive account, Nonlinear differential equations (Granada, 1984) Res. Notes in Math., vol. 132, Pitman, Boston, MA, 1985, pp. 64–113. MR 908899
- J. M. Sanz-Serna and C. Palencia, A general equivalence theorem in the theory of discretization methods, Math. Comp. 45 (1985), no. 171, 143–152. MR 790648, DOI 10.1090/S0025-5718-1985-0790648-7
- J. M. Sanz-Serna and J. G. Verwer, Stability and convergence at the PDE/stiff ODE interface, Appl. Numer. Math. 5 (1989), no. 1-2, 117–132. Recent theoretical results in numerical ordinary differential equations. MR 979551, DOI 10.1016/0168-9274(89)90028-7
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190 V. Thomée, Finite difference methods for linear parabolic equations, Handbook for Numerical Analysis, vol. I. (P. G. Ciarlet and J. L. Lions, eds.), North-Holland, Amsterdam, 1990.
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Math. Comp. 62 (1994), 93-103
- MSC: Primary 47D06; Secondary 34G10, 65J10, 65L99, 65M12
- DOI: https://doi.org/10.1090/S0025-5718-1994-1201070-9
- MathSciNet review: 1201070