2,665 research outputs found
Extrapolation-Based Implicit-Explicit Peer Methods with Optimised Stability Regions
In this paper we investigate a new class of implicit-explicit (IMEX) two-step
methods of Peer type for systems of ordinary differential equations with both
non-stiff and stiff parts included in the source term. An extrapolation
approach based on already computed stage values is applied to construct IMEX
methods with favourable stability properties. Optimised IMEX-Peer methods of
order p = 2, 3, 4, are given as result of a search algorithm carefully designed
to balance the size of the stability regions and the extrapolation errors.
Numerical experiments and a comparison to other implicit-explicit methods are
included.Comment: 21 pages, 6 figure
Linear multistep methods for integrating reversible differential equations
This paper studies multistep methods for the integration of reversible
dynamical systems, with particular emphasis on the planar Kepler problem. It
has previously been shown by Cano & Sanz-Serna that reversible linear
multisteps for first-order differential equations are generally unstable. Here,
we report on a subset of these methods -- the zero-growth methods -- that evade
these instabilities. We provide an algorithm for identifying these rare
methods. We find and study all zero-growth, reversible multisteps with six or
fewer steps. This select group includes two well-known second-order multisteps
(the trapezoidal and explicit midpoint methods), as well as three new
fourth-order multisteps -- one of which is explicit. Variable timesteps can be
readily implemented without spoiling the reversibility. Tests on Keplerian
orbits show that these new reversible multisteps work well on orbits with low
or moderate eccentricity, although at least 100 steps/radian are required for
stability.Comment: 31 pages, 9 figures, in press at The Astronomical Journa
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