An application of Taylor models to the Nakao method on ODEs

The authors give short survey on validated computation of initial value problems for ODEs especially Taylor model methods. Then they propose an application of Taylor models to the Nakao method which has been developed for numerical verification methods on PDEs and apply it to initial value problems for ODEs with some numerical experiments.

[1]  P. Eijgenraam The Solution of Initial Value Problems Using Interval Arithmetic , 1981 .

[2]  Martin Berz,et al.  5. Remainder Differential Algebras and Their Applications , 1996 .

[3]  Wolfgang Kuehn,et al.  Rigorously computed orbits of dynamical systems without the wrapping effect , 1998, Computing.

[4]  Martin Berz,et al.  Verified High-Order Integration of DAEs and Higher-Order ODEs , 2001 .

[5]  Martin Berz,et al.  COSY INFINITY Version 9 , 2006 .

[6]  Rudolf J. Lohner,et al.  On the Ubiquity of the Wrapping Effect in the Computation of Error Bounds , 2001, Perspectives on Enclosure Methods.

[7]  Mitsuhiro T. Nakao NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS , 2000 .

[8]  Kyoko Makino,et al.  Rigorous analysis of nonlinear motion in particle accelerators , 1998 .

[9]  Martin Berz,et al.  Efficient Control of the Dependency Problem Based on Taylor Model Methods , 1999, Reliab. Comput..

[10]  Nathalie Revol,et al.  Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY , 2005, J. Log. Algebraic Methods Program..

[11]  M. Berz,et al.  TAYLOR MODELS AND OTHER VALIDATED FUNCTIONAL INCLUSION METHODS , 2003 .

[12]  Martin Berz,et al.  SUPPRESSION OF THE WRAPPING EFFECT BY TAYLOR MODEL-BASED VERIFIED INTEGRATORS : LONG-TERM STABILIZATION BY SHRINK WRAPPING , 2006 .

[13]  Martin Berz,et al.  SUPPRESSION OF THE WRAPPING EFFECT BY TAYLOR MODEL- BASED VERIFIED INTEGRATORS: LONG-TERM STABILIZATION BY PRECONDITIONING , 2011 .

[14]  A. Neumaier The Wrapping Effect, Ellipsoid Arithmetic, Stability and Confidence Regions , 1993 .