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Verified High-Order Integration of DAEs and Higher-order ODEs

Abstract

Within the framework of Taylor models, no fundamental difference exists between the antiderivation and the more standard elementary operations. Indeed, a Taylor model for the antiderivative of another Taylor model is straightforward to compute and trivially satisfies inclusion monotonicity.

This observation leads to the possibility of treating implicit ODEs and, more importantly, DAEs within a fully Differential Algebraic context, i.e. as implicit equations made of conventional functions as well as the antiderivation. To this end, the highest derivative of the solution function occurring in either the ODE or the constraint conditions of the DAE is represented by a Taylor model. All occurring lower derivatives are represented as antiderivatives of this Taylor model.

By rewriting this derivative-free system in a fixed point form, the solution can be obtained from a contracting Differential Algebraic operator in a finite number of steps. Using Schauder's Theorem, an additional verification step guarantees containment of the exact solution in the computed Taylor model. As a by-product, we obtain direct methods for the integration of higher order ODEs. The performance of the method is illustrated through examples.

J. Hoefkens, M. Berz, K. Makino, in "Scientific Computing, Validated Numerics and Interval Methods", W. Kraemer, J. W. v. Gudenberg (Eds.) (2001) Kluwer

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