On the use of stability regions in the numerical analysis of initial value problems
HTML articles powered by AMS MathViewer
- by H. W. J. Lenferink and M. N. Spijker PDF
- Math. Comp. 57 (1991), 221-237 Request permission
Abstract:
This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations. Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size. These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration. The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product). The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.References
- F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Mathematical Society Lecture Note Series, vol. 2, Cambridge University Press, London-New York, 1971. MR 0288583
- F. F. Bonsall and J. Duncan, Numerical ranges. II, London Mathematical Society Lecture Note Series, No. 10, Cambridge University Press, New York-London, 1973. MR 0442682
- F. F. Bonsall and J. Duncan, Numerical ranges, Studies in functional analysis, MAA Stud. Math., vol. 21, Math. Assoc. America, Washington, D.C., 1980, pp. 1–49. MR 589412
- 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
- J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 557543
- Michel Crouzeix, On multistep approximation of semigroups in Banach spaces, Proceedings of the 2nd international conference on computational and applied mathematics (Leuven, 1986), 1987, pp. 25–35. MR 920377, DOI 10.1016/0377-0427(87)90123-3 M. Crouzeix and P. A. Raviart, Approximation d’équations d’évolution linéaires par des méthodes multipas, Etude Numérique des Grands Systèmes (Rencontres IRIA-Novosibirsk 1976), Dunod, Paris, 1976.
- Germund Dahlquist, $G$-stability is equivalent to $A$-stability, BIT 18 (1978), no. 4, 384–401. MR 520750, DOI 10.1007/BF01932018
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- I. M. Glazman and Ju. I. Ljubič, Finite-dimensional linear analysis: a systematic presentation in problem form, The M.I.T. Press, Cambridge, Mass.-London, 1974. Translated from the Russian and edited by G. P. Barker and G. Kuerti. MR 0354718
- 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), no. 7, 1075–1081. MR 577712, DOI 10.1002/nme.1620150708
- P. J. van der Houwen, Construction of integration formulas for initial value problems, North-Holland Series in Applied Mathematics and Mechanics, Vol. 19, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0519726
- Paul J. Kelly and Max L. Weiss, Geometry and convexity, Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 1979. A study in mathematical methods. MR 534615 J. F. B. M. Kraaijevanger, Private communciation, 1986.
- 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, Proceedings of the 2nd international conference on computational and applied mathematics (Leuven, 1986), 1987, pp. 67–81. MR 920379, DOI 10.1016/0377-0427(87)90126-9
- H. W. J. Lenferink and M. N. Spijker, A generalization of the numerical range of a matrix, Linear Algebra Appl. 140 (1990), 251–266. MR 1075553, DOI 10.1016/0024-3795(90)90232-2
- H. W. J. Lenferink and M. N. Spijker, On a generalization of the resolvent condition in the Kreiss matrix theorem, Math. Comp. 57 (1991), no. 195, 211–220. MR 1079025, DOI 10.1090/S0025-5718-1991-1079025-4
- K. W. Morton, Stability of finite difference approximations to a diffusion-convection equation, Internat. J. Numer. Methods Engrg. 15 (1980), no. 5, 677–683. MR 580354, DOI 10.1002/nme.1620150505
- Olavi Nevanlinna, On the numerical integration of nonlinear initial value problems by linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), no. 1, 58–71. MR 494953, DOI 10.1007/bf01932399 —, Remarks on time discretization of contraction semigroups, Report-HTKK-MAT-A225, Helsinki Univ. Techn., Inst. Math., 1984.
- Nicholas Nirschl and Hans Schneider, The Bauer fields of values of a matrix, Numer. Math. 6 (1964), 355–365. MR 176599, DOI 10.1007/BF01386085
- Seymour V. Parter, Stability, convergence, and pseudo-stability of finite-difference equations for an over-determined problem, Numer. Math. 4 (1962), 277–292. MR 148232, DOI 10.1007/BF01386319
- Satish C. Reddy and Lloyd N. Trefethen, Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues, Comput. Methods Appl. Mech. Engrg. 80 (1990), no. 1-3, 147–164. Spectral and high order methods for partial differential equations (Como, 1989). MR 1067947, DOI 10.1016/0045-7825(90)90019-I
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- Marie-Noëlle Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), no. 147, 919–931. MR 528047, DOI 10.1090/S0025-5718-1979-0528047-2
- Gustaf Söderlind, Bounds on nonlinear operators in finite-dimensional Banach spaces, Numer. Math. 50 (1986), no. 1, 27–44. MR 864303, DOI 10.1007/BF01389666
- M. N. Spijker, Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems, Math. Comp. 45 (1985), no. 172, 377–392. MR 804930, DOI 10.1090/S0025-5718-1985-0804930-8
- Lloyd N. Trefethen, Lax-stability vs. eigenvalue stability of spectral methods, Numerical methods for fluid dynamics, III (Oxford, 1988) Inst. Math. Appl. Conf. Ser. New Ser., vol. 17, Oxford Univ. Press, New York, 1988, pp. 237–253. MR 989156
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 57 (1991), 221-237
- MSC: Primary 65L20; Secondary 65M12
- DOI: https://doi.org/10.1090/S0025-5718-1991-1079026-6
- MathSciNet review: 1079026