288 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
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
Geometrically derived difference formulae for the numerical integration of trajectory problems
The term 'trajectory problem' is taken to include problems that can arise, for instance, in connection with contour plotting, or in the application of continuation methods, or during phase-plane analysis. Geometrical techniques are used to construct difference methods for these problems to produce in turn explicit and implicit circularly exact formulae. Based on these formulae, a predictor-corrector method is derived which, when compared with a closely related standard method, shows improved performance. It is found that this latter method produces spurious limit cycles, and this behavior is partly analyzed. Finally, a simple variable-step algorithm is constructed and tested
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