Digital filters in adaptive time-stepping

Authors:
Gustaf Söderlind
Affiliations:
Lund University, Lund, Sweden
Venue:
ACM Transactions on Mathematical Software (TOMS)
Year:
2003

Citing 11
Cited 21

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Abstract

Adaptive time-stepping based on linear digital control theory has several advantages: the algorithms can be analyzed in terms of stability and adaptivity, and they can be designed to produce smoother stepsize sequences resulting in significantly improved regularity and computational stability. Here, we extend this approach by viewing the closed-loop transfer map Hϕ : logϕ ↦ log h as a digital filter, processing the signal logϕ (the principal error function) in the frequency domain, in order to produce a smooth stepsize sequence log h. The theory covers all previously considered control structures and offers new possibilities to construct stepsize selection algorithms in the asymptotic stepsize-error regime. Without incurring extra computational costs, the controllers can be designed for special purposes such as higher order of adaptivity (for smooth ODE problems) or a stronger ability to suppress high-frequency error components (nonsmooth problems, stochastic ODEs). Simulations verify the controllers' ability to produce stepsize sequences resulting in improved regularity and computational stability.