Difference Methods for Differential Inclusions: A Survey

A. Dontchev, F. Lempio: Difference Methods for Differential Inclusions: A Survey
SIAM Review 34 (2), 263 - 294, 1992

DOI: 10.1137/1034050
MR Nummer: 93g:34024
Zentralblattnummer: 0757.34018
Keywords: convergence of numerical methods; difference equations; difference methods; differential inclusions; mathematical techniques; oscillating systems; vehicle dynamics
Mathematics Subject Classification Code: 34A60 (34A50 49J24 65L05)


Abstract:

The main objective of this survey is to study 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 strate- gies 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.

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

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