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Model Problems in Numerical Stability Theory for Initial Value Problems

Stuart, A. M. and Humphries, A. R. (1994) Model Problems in Numerical Stability Theory for Initial Value Problems. SIAM Review, 36 (2). pp. 226-257. ISSN 0036-1445.

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In the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with simple dynamics; this has been motivated by the need to study error propagation mechanisms in stiff problems, a question modeled effectively by contractive linear or nonlinear problems. While this has resulted in a coherent and self-contained body of knowledge, it has never been entirely clear to what extent this theory is relevant for problems exhibiting more complicated dynamics. 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, in particular, striking similarities between this new developing stability theory and the classical linear and nonlinear stability theories are emphasized. The classical theories of A, B and algebraic stability for Runge–Kutta methods are briefly reviewed; the dynamics of solutions within the classes of equations to which these theories apply—linear decay and contractive problems—are studied. 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 chaotic solutions, are highlighted. Runge-Kutta schemes that 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. It should be emphasized that in all cases the class of methods for which a coherent and complete numerical stability theory exists, given a structural assumption on the initial value problem, is often considerably smaller than the class of methods found to be effective in practice. Nonetheless it is arguable that it is valuable to develop such stability theories to provide a firm theoretical framework in which to interpret existing methods and to formulate goals in the construction of new methods. Furthermore, there are indications that the theory of algebraic stability may sometimes be useful in the analysis of error control codes which are not stable in a fixed step implementation; this work is described.

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Additional Information:© 1994 Society for Industrial and Applied Mathematics. Submitted: 09 November 1992. Accepted: 18 January 1994. The work of this author was supported by Office of Naval Research under N00014-92-J-1876 and National Science Foundation grant DMS-9201727. We are grateful to Luca Deici, Kjell Gustafsson, Arieh Iserles, Bob Russell, Juan Simo, and Marc Spijker for helpful conversations and to the referees for many useful suggestions.
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-92-J-1876
Subject Keywords:numerical stability, Runge–Kutta methods, linear decay, contractivity, gradient systems, dissipativity, conservative systems, Hamiltonian systems
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ27
Issue or Number:2
Classification Code:AMS subject classifications. 34C35, 34D05, 65L07, 65L20
Record Number:CaltechAUTHORS:20170613-100013806
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Official Citation:Model Problems in Numerical Stability Theory for Initial Value Problems A. M. Stuart and A. R. Humphries SIAM Review 1994 36:2, 226-257
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78157
Deposited By: Ruth Sustaita
Deposited On:13 Jun 2017 17:27
Last Modified:03 Oct 2019 18:05

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