Model Problems in Numerical Stability Theory for Initial Value Problems

In the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics. Whilst this has resulted in a coherent and self-contained body of knowledge, it has not thoroughly addressed the problems of real interest in applications. Recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics. This on-going work is unified and possible directions for future work are outlined. In particular, striking similarities between this new developing stability theory and the classical non-linear stability theory are emphasised. The classical theories of $A$, $B$, and algebraic stability for Runge-Kutta methods are briefly reviewed, and it is emphasised that the classes of equations to which these theories apply - linear decay and contractive problems - only admit trivial dynamics. Four other categories of equations - gradient, dissipative, conservative and Hamiltonian systems - are considered. Relationships and differences between the possible dynamics in each category, which range from multiple competing equilibria to fully chaotic solutions, are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications. Runge-Kutta schemes which preserve the dynamical structure of the underlying problem are sought, and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given. Algebraic stability, in particular, is seen to play a central role. The effects of error control are considered, and multi-step methods are discussed briefly. Finally, various open problems are described.

[1]  R. Jeltsch,et al.  Generalized disks of contractivity for explicit and implicit Runge-Kutta methods , 1979 .

[2]  R. Russell,et al.  Unitary integrators and applications to continuous orthonormalization techniques , 1994 .

[3]  R. Nicolaides,et al.  Numerical analysis of a continuum model of phase transition , 1991 .

[4]  M. N. Spijker.  Contractivity in the numerical solution of initial value problems , 1983 .

[5]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[6]  G. P. Szegö,et al.  Stability theory of dynamical systems , 1970 .

[7]  Andrew M. Stuart,et al.  Numerical analysis of dynamical systems , 1994, Acta Numerica.

[8]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[9]  Andrew M. Stuart,et al.  Approximation of dissipative partial differential equations over long time intervals , 1994 .

[10]  Germund Dahlquist,et al.  G-stability is equivalent toA-stability , 1978 .

[11]  G. J. Cooper.  Stability of Runge-Kutta Methods for Trajectory Problems , 1987 .

[12]  J. Butcher.  Implicit Runge-Kutta processes , 1964 .

[13]  A. Iserles.  Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations , 1990 .

[14]  J. M. Sanz-Serna,et al.  Equilibria of Runge-Kutta methods , 1990 .

[15]  J. M. Sanz-Serna,et al.  Symplectic integrators for Hamiltonian problems: an overview , 1992, Acta Numerica.

[16]  G. Dahlquist.  Error analysis for a class of methods for stiff non-linear initial value problems , 1976 .

[17]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[18]  Andrew M. Stuart.  Linear Instability Implies Spurious Periodic Solutions , 1989 .

[19]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[20]  Andrew M. Stuart,et al.  Runge-Kutta methods for dissipative and gradient dynamical systems , 1994 .

[21]  Edriss S. Titi,et al.  Dissipativity of numerical schemes , 1991 .

[22]  John C. Butcher,et al.  A stability property of implicit Runge-Kutta methods , 1975 .

[23]  Andrew M. Stuart,et al.  The essential stability of local error control for dynamical systems , 1995 .

[24]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[25]  Donald A. French,et al.  Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems , 1994 .

[26]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[27]  George Hall,et al.  Equilibrium states of Runge Kutta schemes , 1985, TOMS.

[28]  M. N. Spijker,et al.  A note onB-stability of Runge-Kutta methods , 1980 .

[29]  J. Kraaijevanger.  Contractivity of Runge-Kutta methods , 1991 .

[30]  J. M. Sanz-Serna,et al.  Runge-kutta schemes for Hamiltonian systems , 1988 .

[31]  G. J. Cooper.  Weak Nonlinear Stability of Implicit Runge-Kutta Methods , 1992 .

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

[33]  Mari Paz Calvo,et al.  The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem , 1993, SIAM J. Sci. Comput..

[34]  M. N. Spijker.  A note on contractivity in the numerical solution of initial value problems , 1987 .

[35]  Kevin Burrage,et al.  High order algebraically stable Runge-Kutta methods , 1978 .

[36]  M. Crouzeix.  Sur laB-stabilité des méthodes de Runge-Kutta , 1979 .

[37]  F. Lasagni.  Canonical Runge-Kutta methods , 1988 .