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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ascher, U. M. and Petzold, L. R. (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM.
Berz, M. (1999). Modern Map Methods in Particle Beam Physics. Academic Press, San Diego.
Berz, M. (2000). Higher order verified methods and applications. SCAN2000, Kluver.
Berz, M. and Hoefkens, J. (2001). Verfied inversion of functional dependencies and superconvergent interval Newton methods. Reliable Computing, 7(5).
Berz, M., Hoffstätter, G., Wan, W., Shamseddine, K., and Makino, K. (1996). COSY INFINITY and its applications to nonlinear dynamics. In Berz, M., Bischof, C., Corliss, G., and Griewank, A., editors, Computational Differentiation: Techniques, Applications, and Tools, pages 363–365, Philadelphia. SIAM.
Berz, M. and Makino, K. (1998). Verified integration of ODEs and flows with differential algebraic methods on Taylor models. Reliable Computing, 4: 361–369.
Hoefkens, J. and Berz, M. (2001a). Efficient high-order methods for ODEs and DAEs. Proc. AD2000, SIAM.
Hoefkens, J. and Berz, M. (200 lb). Verification of invertibility of complicated functions over large domains. Reliable Computing.
Lohner, R. J. (1987). Enclosing the solutions of ordinary initial and boundary value problems. In Kaucher, E. W., Kulisch, U. W., and Ullrich, C., editors, Computer Arithmetic: Scientific Computation and Programming Languages, pages 255–286. WileyTeubner Series in Computer Science, Stuttgart.
Makino, K. (1998). Rigorous Analysis of Nonlinear Motion in Particle Accelerators. PhD thesis, Michigan State University, East Lansing, Michigan, USA. also http://bt.nscl.msu.edu/pub and MSUCL-1093.
Makino, K. and Berz, M. (1996). Remainder differential algebras and their applications. In Berz, M., Bischof, C., Corliss, G., and Griewank, A., editors, Computational Differentiation: Techniques, Applications, and Tools, pages 63–74, Philadelphia. SIAM.
Makino, K. and Berz, M. (1999). Efficient control of the dependency problem based on Taylor model methods. Reliable Computing, 5: 3–12.
Makino, K. and Berz, M. (2000). Advances in verified integration of ODEs. SCAN2000.
Makino, K., Berz, M., and Hoefkens, J. (2000). Differential algebraic structures and verification. ACA2000.
Nedialkov, N. S., Jackson, K. R., and Corliss, G. E (to appear, 2000). Validated solutions of initial value problems for ordinary differential equations. Appl. Math. & Comp. see also http://www.mscs.mu.edu/georgec/Pubs/journ.html#1999f.
Pantelides, C. C. (1988). The consistent initialization of differential-algebraic systems. SIAM Journal on Scientific and Statistical Computing, 9 (2): 213–231.
Pryce, J. D. (2000). A simple structural analysis method for DAEs. Technical Report DoIS/ TR05/ 00, RMCS, Cranfield University.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Hoefkens, J., Berz, M., Makino, K. (2001). Verified High-Order Integration of DAEs and Higher-Order ODEs. In: Krämer, W., von Gudenberg, J.W. (eds) Scientific Computing, Validated Numerics, Interval Methods. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6484-0_23
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6484-0_23
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-3376-8
Online ISBN: 978-1-4757-6484-0
eBook Packages: Springer Book Archive