Linear stability analysis in the numerical solution of initial value problems

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.

[1]  Fritz John,et al.  On integration of parabolic equations by difference methods: I. Linear and quasi‐linear equations for the infinite interval , 1952 .

[2]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[3]  Difference methods for mixed boundary-value problems , 1960 .

[4]  Seymour V. Parter,et al.  Stability, convergence, and pseudo-stability of finite-difference equations for an over-determined problem , 1962 .

[5]  H. Kreiss.  Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren , 1962 .

[6]  Gilbert Strang,et al.  Accurate partial difference methods , 1964 .

[7]  K. Morton.  On a matrix theorem due to H. O. Kreiss , 1964 .

[8]  G. Strang,et al.  Matrix theorems for partial differential and difference equations , 1965 .

[9]  C. McCarthy,et al.  On the norm of a finite boolean algebra of projections, and applications to theorems of Kreiss and Morton† , 1965 .

[10]  Difference approximations for the initial-boundary value problem for hyperbolic differential equations , 1966 .

[11]  C. Pearcy.  An elementary proof of the power inequality for the numerical radius. , 1966 .

[12]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[13]  D. Greenspan.  Numerical Solutions of Nonlinear Differential Equations , 1967 .

[14]  H. Kreiss.  Stability theory for difference approximations of mixed initial boundary value problems. I , 1968 .

[15]  On the resolvent of a linear operator associated with a well-posed Cauchy problem , 1968 .

[16]  J. Crank.  Difference Methods for Initial-Value Problems 2nd edn , 1968 .

[17]  Harold Widom,et al.  An inequality for rational functions , 1970 .

[18]  J. Duncan,et al.  Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras: Hermitian elements of a complex unital Banach algebra , 1971 .

[19]  M. N. Spijker.  Equivalence theorems for nonlinear finite-difference methods , 1972 .

[20]  H. Kreiss,et al.  Stability Theory of Difference Approximations for Mixed Initial Boundary Value Problems. II , 1972 .

[21]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[22]  H. Landau,et al.  On Szegö’s eingenvalue distribution theorem and non-Hermitian kernels , 1975 .

[23]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[24]  D. Gottlieb,et al.  Numerical analysis of spectral methods , 1977 .

[25]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[26]  Rolf Dieter Grigorieff,et al.  Numerical Treatment of Differential Equations , 1978 .

[27]  V. Thomée,et al.  ON RATIONAL APPROXIMATIONS OF SEMIGROUPS , 1979 .

[28]  J. Varah.  On the Separation of Two Matrices , 1979 .

[29]  K. W. Morton.  Stability of finite difference approximations to a diffusion–convection equation , 1980 .

[30]  Studies in Functional Analysis , 1980 .

[31]  David F. Griffiths,et al.  Analysis of error growth for explicit difference schemes in conduction–convection problems , 1980 .

[32]  E. Tadmor.  The equivalence of L2-stability, the resolvent condition, and strict H-stability , 1981 .

[33]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[34]  L. Trefethen,et al.  On the resolvent condition in the Kreiss Matrix Theorem , 1984 .

[35]  Fritz John,et al.  On Integration of Parabolic Equations By Difference Methods , 1985 .

[36]  Remarks on Time Discretization of Contraction Semigroups , 1985 .

[37]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[38]  M. N. Spijker.  Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems , 1985 .

[39]  Advanced Problems: 6505-6506 , 1985 .

[40]  D. Whittaker,et al.  A Course in Functional Analysis , 1991, The Mathematical Gazette.

[41]  M. Crouzeix.  On multistep approximation of semigroups in Banach spaces , 1987 .

[42]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[43]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[44]  M. N. Spijker,et al.  Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations , 1987 .

[45]  J. M. Sanz-Serna,et al.  Stability and Convergence in Numerical Analysis III: Linear Investigation of Nonlinear Stability , 1988 .

[46]  Vidar Thomée,et al.  Finite difference methods for linear parabolic equations , 1990 .

[47]  Lloyd N. Trefethen,et al.  Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues , 1990 .

[48]  M. N. Spijker,et al.  A generalization of the numerical range of a matrix , 1990 .

[49]  C. Lubich,et al.  On resolvent conditions and stability estimates , 1991 .

[50]  M. N. Spijker,et al.  On a generalization of the resolvent condition in the Kreiss matrix theorem , 1991 .

[51]  M. N. Spijker.  On a conjecture by le Veque and Trefethen related to the kreiss matrix theorem , 1991 .

[52]  M. N. Spijker,et al.  On the use of stability regions in the numerical analysis of initial value problems , 1991 .

[53]  L. Trefethen,et al.  Eigenvalues and pseudo-eigenvalues of Toeplitz matrices , 1992 .

[54]  C. Lubich,et al.  On the convergence of multistep methods for nonlinear stiff differential equations , 1992 .

[55]  L. Trefethen,et al.  Stability of the method of lines , 1992, Spectra and Pseudospectra.

[56]  Stig Larsson,et al.  The stability of rational approximations of analytic semigroups , 1993 .

[57]  H. Kreiss,et al.  On the stability definition of difference approximations for the initial boundary value problem , 1993 .

[58]  Two counterexamples related to the Kreiss matrix theorem , 1994 .

[59]  L. Trefethen,et al.  From the Buffon Needle Problem to the Kreiss Matrix Theorem , 1994 .