2,015 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
Vibrational resonance: a study with high-order word-series averaging
We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the technique applied here, the tasks of constructing the averaged system and the associated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of simple recursions. As distinct from the situation with other approaches, with word-series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approximations to the true dynamics.Ministerio de EconomĆa, Industria y Competitividad, proyectos MTM2013-46553-C3-1-P y MTM2013-46553-C3-2-
From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials
We present a multiscale integrator for Hamiltonian systems with slowly
varying quadratic stiff potentials that uses coarse timesteps (analogous to
what the impulse method uses for constant quadratic stiff potentials). This
method is based on the highly-non-trivial introduction of two efficient
symplectic schemes for exponentiations of matrices that only require O(n)
matrix multiplications operations at each coarse time step for a preset small
number n. The proposed integrator is shown to be (i) uniformly convergent on
positions; (ii) symplectic in both slow and fast variables; (iii) well adapted
to high dimensional systems. Our framework also provides a general method for
iteratively exponentiating a slowly varying sequence of (possibly high
dimensional) matrices in an efficient way
Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position
Impulse methods are generalized to a family of integrators for Langevin
systems with quadratic stiff potentials and arbitrary soft potentials. Uniform
error bounds (independent from stiff parameters) are obtained on integrated
positions allowing for coarse integration steps. The resulting integrators are
explicit and structure preserving (quasi-symplectic for Langevin systems)
Exterior complex scaling as a perfect absorber in time-dependent problems
It is shown that exterior complex scaling provides for complete absorption of
outgoing flux in numerical solutions of the time-dependent Schr\"odinger
equation with strong infrared fields. This is demonstrated by computing high
harmonic spectra and wave-function overlaps with the exact solution for a
one-dimensional model system and by three-dimensional calculations for the H
atom and a Ne atom model. We lay out the key ingredients for correct
implementation and identify criteria for efficient discretization
A stroboscopic averaging algorithm for highly oscillatory delay problems
We propose and analyse a heterogeneous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method suggested here may provide approximations with O(H2+1/Ī©2) errors with a computational effort that grows like Hā1 (the inverse of the step size), uniformly in the forcing frequency Ī©ā .J.M. Sanz-Serna has been supported by projects MTM2013-46553-C3-1-P from Ministerio de EconomĆa y Comercio, and MTM2016-77660-P(AEI/FEDER, UE) from Ministerio de EconomĆa, Industria y Competitividad, Spain. Beibei Zhu has been supported by the National Natural Science Foundation of China (Grant No. 11371357 and No. 11771438). She is grateful to Universidad Carlos III de Madrid for hosting the stay in Spain that made this work possible and to the Chinese Scholarship Council for providing the necessary funds. The authors are thankful to M. A. F. SanjuĆ”n and A. Daza for bringing to their attention the vibrational resonance phenomenon, the toggle switch problem and other highly-oscillatory systems with delay
Spectral methods in fluid dynamics
Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
- ā¦