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Some recent advances in validated methods for IVPs for ODEs. (English) Zbl 0998.65068

Summary: Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced.
We present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and discuss some recent advances in the theory of validated methods for IVPs for ODEs. In particular, we discuss an interval Hermite-Obreschkoff (IHO) scheme for computing rigorous bounds on the solution of an IVP for an ODE, the stability of ITS and IHO methods, and a new perspective on the wrapping effect, where we interpret the problem of reducing the wrapping effect as one of finding a more stable scheme for advancing the solution.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65G20 Algorithms with automatic result verification
65G40 General methods in interval analysis
34A34 Nonlinear ordinary differential equations and systems
65L70 Error bounds for numerical methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations

Software:

FADBAD++; TADIFF
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References:

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