Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-12T22:47:40.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear stability analysis in the numerical solution of initial value problems

Published online by Cambridge University Press:  07 November 2008

J.L.M. van Dorsselaer ,
J.F.B.M. Kraaijevanger  and
M.N. Spijker
J.L.M. van Dorsselaer
Affiliation:
Department of Mathematics and Computer Science, University of LeidenThe Netherlands E-mail: [email protected]
J.F.B.M. Kraaijevanger
Affiliation:
Department of Mathematics and Computer Science, University of LeidenThe Netherlands E-mail: [email protected]
M.N. Spijker
Affiliation:
Department of Mathematics and Computer Science, University of LeidenThe Netherlands E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

This article addresses the general problem of establishing upper bounds for the norms of the nth powers of square matrices. The focus is on upper bounds that grow only moderately (or stay constant) where n, or the order of the matrices, increases. The so-called resolvant condition, occuring in the famous Kreiss matrix theorem, is a classical tool for deriving such bounds.

Recently the classical upper bounds known to be valid under Kreiss's resolvant condition have been improved. Moreover, generalizations of this resolvant condition have been considered so as to widen the range of applications. The main purpose of this article is to review and extend some of these new developments.

The upper bounds for the powers of matrices discussed in this article are intimately connected with the stability analysis of numerical processes for solving initial(-boundary) value problems in ordinary and partial linear differential equations. The article highlights this connection.

The article concludes with numerical illustrations in the solution of a simple initial-boundary value problem for a partial differential equation.

Type
Research Article
Information
Acta Numerica , Volume 2 , January 1993 , pp. 199 - 237
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. and Stegun, I.A. (1965), Handbook of Mathematical Functions, Dover (New York).Google Scholar
Bonsall, F.F. and Duncan, J. (1971), Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge University Press (Cambridge).CrossRefGoogle Scholar
Bonsall, F.F. and Duncan, J. (1980), ‘Numerical ranges’, in Studies in Functional Analysis (Bartle, R.G., ed.), The Mathematical Association of America, 1–49.Google Scholar
Brenner, Ph. and Thomée, V. (1979), ‘On rational approximations of semigroups’, SIAM J. Numer. Anal. 16, 683–694.Google Scholar
Butcher, J.C. (1987), The Numerical Analysis of Ordinary Differential Equations, John Wiley (Chichester).Google Scholar
Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988), Spectral Methods in Fluid Dynamics, Springer (New York).CrossRefGoogle Scholar
Conway, J.B. (1985), A Course in Functional Analysis, Springer (New York).CrossRefGoogle Scholar
Crouzeix, M. (1987), ‘On multistep approximation of semigroups in Banach spaces’, J. Comput. Appl. Math. 20, 25–35.Google Scholar
Crouzeix, M., Larsson, S., Piskarev, S. and Thomée, V. (1991), ‘The stability of rational approximations of analytic semigroups’, Technical Report 1991:28, Chalmers University of Technology & the University of Göteborg (Göteborg).Google Scholar
Forsythe, G.E. and Wasow, W.R. (1960), Finite Difference Methods for Partial Differential Equations, John Wiley (New York).Google Scholar
Gottlieb, D. and Orszag, S.A. (1977), Numerical Analysis of Spectral Methods, Soc. Ind. Appl. Math. (Philadelphia).CrossRefGoogle Scholar
Griffiths, D.F., Christie, I. and Mitchell, A.R. (1980), ‘Analysis of error growth for explicit difference schemes in conduction-convection problems’, Int. J. Numer. Meth. Engrg 15, 1075–1081.CrossRefGoogle Scholar
Grigorieff, R.D. (1991), ‘Time discretization of semigroups by the variable two-step BDF method’, in Numerical Treatment of Differential Equations (Strehmel, K., ed.), Teubner (Leipzig), 204–216.Google Scholar
Gustafsson, B., Kreiss, H.-O. and Sundström, A. (1972), ‘Stability theory of difference approximations for mixed initial boundary value problems. II’, Math. Comput. 26, 649–686.CrossRefGoogle Scholar
Hairer, E. and Wanner, G. (1991), Solving Ordinary Differential Equations, Vol. II, Springer (Berlin).Google Scholar
Henrici, P. (1974), Applied and Computational Complex Analysis, Vol. 1, John Wiley (New York).Google Scholar
Horn, R.A. and Johnson, C.R. (1990), Matrix Analysis, Cambridge University Press (Cambridge).Google Scholar
John, F. (1952), ‘On integration of parabolic equations by difference methods’, Comm. Pure Appl. Math. 5, 155–211.CrossRefGoogle Scholar
Kraaijevanger, J.F.B.M. (1992), ‘Two counterexamples related to the Kreiss matrix theorem’, submitted to BIT.Google Scholar
Kraaijevanger, J.F.B.M., Lenferink, H.W.J. and Spijker, M.N. (1987), ‘Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations’, J. Comput. Appl. Math. 20, 67–81.CrossRefGoogle Scholar
Kreiss, H.-O. (1962), ‘Über die Stabilitätsdefinition fü r Differenzengleichungen die partielle Differentialgleichungen approximieren’, BIT 2, 153–181.Google Scholar
Kreiss, H.-O. (1966), ‘Difference approximations for the initial-boundary value problem for hyperbolic differential equations’, in Numerical Solution of Nonlinear Differential Equations (Greenspan, D., ed.), John Wiley (New York), 141–166.Google Scholar
Kreiss, H.-O. (1990), ‘Well posed hyperbolic initial boundary value problems and stable difference approximations’, in Proc. Third Int. Conf. on Hyperbolic Problems, Uppsala, Sweden.Google Scholar
Kreiss, H.-O. and Wu, L. (1992), ‘On the stability definition of difference approximations for the initial boundary value problem’, to appear in Comm. Pure Appl. Math.Google Scholar
Landau, H.J. (1975), ‘On Szegő 's eigenvalue distribution theorem and non-Hermitian kernels’, J. d' Analyse Math. 28, 335–357.Google Scholar
Laptev, G.I. (1975), ‘Conditions for the uniform well-posedness of the Cauchy problem for systems of equations’, Sov. Math. Dokl. 16, 65–69.Google Scholar
Lenferink, H.W.J. and Spijker, M.N. (1990), ‘A generalization of the numerical range of a matrix’, Linear Algebra Appl. 140, 251–266.Google Scholar
Lenferink, H.W.J. and Spijker, M.N. (1991a), ‘On a generalization of the resolvent condition in the Kreiss matrix theorem’, Math. Comput. 57, 211–220.CrossRefGoogle Scholar
Lenferink, H.W.J. and Spijker, M.N. (1991b), ‘On the use of stability regions in the numerical analysis of initial value problems’, Math. Comput. 57, 221–237.Google Scholar
LeVeque, R.J. and Trefethen, L.N. (1984), ‘On the resolvent condition in the Kreiss matrix theorem’, BIT 24, 584–591.CrossRefGoogle Scholar
López-Marcos, J.C. and Sanz-Serna, J.M. (1988), ‘Stability and convergence in numerical analysis III: linear investigation of nonlinear stability’, IMA J. Numer. Anal. 8, 71–84.CrossRefGoogle Scholar
Lubich, C. (1991), ‘On the convergence of multistep methods for nonlinear stiff differential equations’, Numer. Math. 58, 839–853.Google Scholar
Lubich, C. and Nevanlinna, O. (1991), ‘On resolvent conditions and stability estimates’, BIT 31, 293–313.CrossRefGoogle Scholar
McCarthy, C.A. (1992), ‘A strong resolvent condition need not imply power-boundedness’, to appear in J. Math. Anal. Appl.Google Scholar
McCarthy, C.A. and Schwartz, J. (1965), ‘On the norm of a finite boolean algebra of projections, and applications to theorems of Kreiss and Morton’, Comm. Pure Appl. Math. 18, 191–201.Google Scholar
Meis, T. and Marcowitz, U. (1981), Numerical Solution of Partial Differential Equations, Springer (New York).Google Scholar
Miller, J. (1968), ‘On the resolvent of a linear operator associated with a well-posed Cauchy problem’, Math. Comput. 22, 541–548.Google Scholar
Miller, J. and Strang, G. (1966), ‘Matrix theorems for partial differential and difference equations’, Math. Scand. 18, 113–123.CrossRefGoogle Scholar
Morton, K.W. (1964), ‘On a matrix theorem due to H.O. Kreiss’, Comm. Pure Appl. Math. 17, 375–379.Google Scholar
Morton, K.W. (1980), ‘Stability of finite difference approximations to a diffusion-convection equation’, Int. J. Num. Meth. Engrg 15, 677–683.Google Scholar
Nevanlinna, O. (1984), ‘Remarks on time discretization of contraction semigroups’, Report HTKK-MAT-A225, Helsinki Univ. Techn. (Helsinki).Google Scholar
Oden, J.T. and Reddy, J.N. (1976), An Introduction to the Mathematical Theory of Finite Elements, John Wiley (New York).Google Scholar
Parter, S.V. (1962), ‘Stability, convergence, and pseudo-stability of finite-difference equations for an over-determined problem’, Numer. Math. 4, 277–292.CrossRefGoogle Scholar
Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer (New York).Google Scholar
Pearcy, C. (1966), ‘An elementary proof of the power inequality for the numerical radius’, Michigan Math. J. 13, 289–291.CrossRefGoogle Scholar
Reddy, S.C. and Trefethen, L.N. (1990), ‘Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues’, Comput. Meth. Appl. Mech. Engrg 80, 147–164.CrossRefGoogle Scholar
Reddy, S.C. and Trefethen, L.N. (1992), ‘Stability of the method of lines’, Numer. Math. 62, 235–267.Google Scholar
Reichel, L. and Trefethen, L.N. (1992), ‘Eigenvalues and pseudo-eigenvalues of Toeplitz matrices’, Linear Algebra Appl. 162–164, 153–185.Google Scholar
Richtmyer, R.D. and Morton, K.W. (1967), Difference Methods for Initial-value Problems, 2nd Ed., John Wiley (New York).Google Scholar
Rudin, W. (1973), Functional Analysis, McGraw-Hill (New York).Google Scholar
Smith, J.C. (1985), ‘An inequality for rational functions’, Amer. Math. Monthly 92, 740–741.Google Scholar
Spijker, M.N. (1972), ‘Equivalence theorems for nonlinear finite-difference methods’, Springer Lecture Notes in Mathematics, Vol. 267, Springer (Berlin), 233–264.Google Scholar
Spijker, M.N. (1985), ‘Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems’, Math. Comput. 45, 377–392.CrossRefGoogle Scholar
Spijker, M.N. (1991), ‘On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem’, BIT 31, 551–555.Google Scholar
Strang, W.G. (1960), ‘Difference methods for mixed boundary-value problems’, Duke Math. J. 27, 221–231.CrossRefGoogle Scholar
Strang, G. (1964), ‘Accurate partial difference methods II. Non-linear problems’, Numer. Math. 6, 37–46.Google Scholar
Strang, G. and Fix, G.J. (1973), An Analysis of the Finite Element Method, Prentice-Hall (Englewood Cliffs).Google Scholar
Tadmor, E. (1981), ‘The equivalence of L2-stability, the resolvent condition, and strict H-stability’, Linear Algebra Appl. 41, 151–159.Google Scholar
Thomée, V. (1990), ‘Finite difference methods for linear parabolic equations’, in Handbook of Numerical Analysis I (Ciarlet, P.G. and Lions, J.L., eds), North-Holland (Amsterdam), 5–196.CrossRefGoogle Scholar
Trefethen, L.N. (1988), ‘Lax-stability vs. eigenvalue stability of spectral methods’, in Numerical Methods for Fluid Dynamics III (Morton, K.W. and Baines, M.J., eds), Clarendon Press (Oxford), 237–253.Google Scholar
Trefethen, L.N. (1992), Spectra and Pseudospectra, book in preparation.Google Scholar
Varah, J.M. (1979), ‘On the separation of two matrices’, SIAM J. Numer. Anal. 16, 216–222.CrossRefGoogle Scholar
Wegert, E. and Trefethen, L.N. (1992), ‘From the Buffon needle problem to the Kreiss matrix theorem’, to appear in Amer. Math. Monthly.Google Scholar