Convergence and Stability in the Numerical Approximation of Dynamical Systems

1 Outline In this article we give an overview of the application of theories from dynami-cal systems to the analysis of numerical methods for initial-value problems. We start by describing the classical viewpoints of numerical analysis and of dynami-cal systems and then indicate how the two viewpoints can be merged to provide a framework for both the interpretation of data obtained from numerical simulations and the design of eeecient numerical methods. This is done in section 2. In addressing the question of how to interpret data, we will show in section 3 how the concept of convergence can be generalized to the numerical approximation of dynamical systems. The main theory is developed for one-step methods for ordinary diierential equations and extensions to the study of multistep methods , adaptive time-stepping algorithms and partial diierential equations are then outlined. In addressing the question of designing eeecient schemes we will show in section 4 how the concept of stability can be generalized to dynamical systems. Stability theory is developed for both one-step and multistep methods for ordinary diierential equations; extensions to adaptive time-stepping and to partial diierential equations are also outlined. A variety of surveys of this eld already exist { see 13] for a complete study, see 9] for a discussion of convergence in the dynamical systems context, see 12] for a discussion of stability in the dynamical systems context and see 5] for a discussion of both issues in relation to discretization of partial diierential equations. Since these surveys contain exhaustive bibliographys we refer to them for detailed references.