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Stepsize selection in the rigorous defect control of Taylor series methods. (English) Zbl 1429.65145

Summary: Modern numerical methods for initial-value problems (IVPs) in ordinary differential equations (ODE) produce a (piecewise) differentiable numerical solution. The defect or residual evaluated at it induces a perturbed ODE that is satisfied exactly by this solution. Defect control methods try to ensure that a norm of the defect is bounded by a user-specified tolerance. This paper justifies and implements a simple and effective stepsize selection strategy and an overall method for both validated and non-validated defect control of an explicit Taylor series method for IVP ODEs. In validated mode, this method guarantees that the defect is bounded by a user-specified tolerance.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
65G20 Algorithms with automatic result verification
65G40 General methods in interval analysis
41A10 Approximation by polynomials

Software:

FADBAD++; NSDTST; STDTST
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References:

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