Contractivity-preserving implicit linear multistep methods

We investigate contractivity properties of implicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on nonlinear and linear systems d U(t) = f(t, U(t)), where f satisfies a so-called circle condition in an arbitrary norm. The results for the two types of systems turn out to be closely related. We construct optimal multistep methods of given order and stepnumber, which allow the use of a maximal stepsize.

[1]  F. Gantmacher,et al.  Applications of the theory of matrices , 1960 .

[2]  P. Henrici.  Discrete Variable Methods in Ordinary Differential Equations , 1962 .

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

[4]  Walter Gautschi,et al.  On inverses of Vandermonde and confluent Vandermonde matrices. II , 1963 .

[5]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[6]  C. Bolley,et al.  Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques , 1978 .

[7]  Werner Liniger,et al.  Contractive methods for stiff differential equations Part II , 1978 .

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

[9]  O. Nevanlinna,et al.  Stability of explicit time discretizations for solving initial value problems , 1982 .

[10]  Reiner Vanselow,et al.  Nonlinear stability behaviour of linear multistep methods , 1983 .

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

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

[13]  J. Kraaijevanger,et al.  Absolute monotonicity of rational functions occurring in the numerical solution of initial value problems , 1986 .

[14]  Jörgen Sand.  Circle contractive linear multistep methods , 1986 .

[15]  J. Kraaijevanger,et al.  Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems , 1986 .

[16]  John C. Butcher,et al.  The equivalence of algebraic stability andAN-stability , 1987 .

[17]  Choices in contractivity , 1989 .

[18]  H. Lenferink,et al.  Contractivity preserving explicit linear multistep methods , 1989 .