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Asen Dontchev, Frank Lempio

Difference Methods for Differential Inclusions: A Survey

The paper is published in:
SIAM Review 34 (1992), no. 2, pp. 263-294.
34A60 Equations with multivalued right-hand sides, See Also
49M25 Finite difference methods
65L05 Initial value problems

differential inclusions, difference methods

The main objective of this survey is the study of convergence properties of difference methods applied to differential
inclusions. It presents, in a unified way, a number of results scattered in the literature and provides also an introduction to
the topic.

Convergence proofs for the classical Euler method and for a class of multistep methods are outlined. It is shown how
numerical methods for stiff differential equations can be adapted to differential inclusions with additional monotonicity
properties. Together with suitable localization procedures this approach results in higher order methods.

Convergence properties of difference methods with selection strategies are investigated, especially strategies forcing
convergence to solutions with additional smoothness properties.

The error of the Euler method, represented by the Hausdorff distance between the set of approximate solutions and the set
of exact solutions is estimated. First and second order approximations to the reachable sets are presented.

Table of Contents:
1. Introduction
2. Euler Method
3. Convergent Multistep Methods
4. One-Sided Lipschitz Condition and Monotonicity
5. Selection Strategies
6. Error Estimates
7. Convergence of Reachable Sets
8. Higher Order Approximations to Reachable Sets
9. Concluding Remarks

© Robert Baier
Last modified: Thu May 28 15:23:12 MDT 1998