28,780 research outputs found
Some robust integrators for large time dynamics
This article reviews some integrators particularly suitable for the numerical
resolution of differential equations on a large time interval. Symplectic
integrators are presented. Their stability on exponentially large time is shown
through numerical examples. Next, Dirac integrators for constrained systems are
exposed. An application on chaotic dynamics is presented. Lastly, for systems
having no exploitable geometric structure, the Borel-Laplace integrator is
presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are
carried out, as well as on a partial differential equation.
Keywords: Symplectic integrators, Dirac integrators, long-time stability,
Borel summation, divergent series.Comment: 33 pages, 18 figure
Exact Global Control of Small Divisors in Rational Normal Form
Rational normal form is a powerful tool to deal with Hamiltonian partial
differential equations without external parameters. In this paper, we build
rational normal form with exact global control of small divisors. As an
application to nonlinear Schr\"{o}dinger equations in Gevrey spaces, we prove
sub-exponentially long time stability results for generic small initial data
Hamiltonian reduction using a convolutional auto-encoder coupled to an Hamiltonian neural network
The reduction of Hamiltonian systems aims to build smaller reduced models,
valid over a certain range of time and parameters, in order to reduce computing
time. By maintaining the Hamiltonian structure in the reduced model, certain
long-term stability properties can be preserved. In this paper, we propose a
non-linear reduction method for models coming from the spatial discretization
of partial differential equations: it is based on convolutional auto-encoders
and Hamiltonian neural networks. Their training is coupled in order to
simultaneously learn the encoder-decoder operators and the reduced dynamics.
Several test cases on non-linear wave dynamics show that the method has better
reduction properties than standard linear Hamiltonian reduction methods.Comment: 29 pages, 15 figure
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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