899 research outputs found
A stroboscopic averaging algorithm for highly oscillatory delay problems
We propose and analyze a heterogenous multiscale method for the efficient
integration of constant-delay differential equations subject to fast periodic
forcing. The stroboscopic averaging method (SAM) suggested here may provide
approximations with \(\mathcal{O}(H^2+1/\Omega^2)\) errors with a
computational effort that grows like \(H^{-1}\) (the inverse of the
stepsize), uniformly in the forcing frequency Omega
Formal series and numerical integrators: some history and some new techniques
This paper provides a brief history of B-series and the associated Butcher
group and presents the new theory of word series and extended word series.
B-series (Hairer and Wanner 1976) are formal series of functions parameterized
by rooted trees. They greatly simplify the study of Runge-Kutta schemes and
other numerical integrators. We examine the problems that led to the
introduction of B-series and survey a number of more recent developments,
including applications outside numerical mathematics. Word series (series of
functions parameterized by words from an alphabet) provide in some cases a very
convenient alternative to B-series. Associated with word series is a group G of
coefficients with a composition rule simpler than the corresponding rule in the
Butcher group. From a more mathematical point of view, integrators, like
Runge-Kutta schemes, that are affine equivariant are represented by elements of
the Butcher group, integrators that are equivariant with respect to arbitrary
changes of variables are represented by elements of the word group G.Comment: arXiv admin note: text overlap with arXiv:1502.0552
Palindromic 3-stage splitting integrators, a roadmap
The implementation of multi-stage splitting integrators is essentially the
same as the implementation of the familiar Strang/Verlet method. Therefore
multi-stage formulas may be easily incorporated into software that now uses the
Strang/Verlet integrator. We study in detail the two-parameter family of
palindromic, three-stage splitting formulas and identify choices of parameters
that may outperform the Strang/Verlet method. One of these choices leads to a
method of effective order four suitable to integrate in time some partial
differential equations. Other choices may be seen as perturbations of the
Strang method that increase efficiency in molecular dynamics simulations and in
Hybrid Monte Carlo sampling.Comment: 20 pages, 8 figures, 2 table
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