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Generalized Shadow Hybrid Monte Carlo (GSHMC) is a method for molecular simulations that rigorously alternates Monte Carlo sampling from a canonical ensemble with integration of trajectories using Molecular Dynamics (MD). While conventional hybrid Monte Carlo methods completely re-sample particle’s velocities between MD trajectories, our method suggests a partial velocity update procedure which keeps a part of the dynamic information throughout the simulation. We use shadow (modified) Hamiltonians,...
The evolution of n–dimensional graphs under a
weighted curvature flow is approximated by linear finite elements. We obtain
optimal error bounds for the normals and the normal velocities of the surfaces
in natural norms.
Furthermore we prove a global existence result for the
continuous problem and present some examples of computed surfaces.
We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.
In this paper, we consider C1,1 Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the SDH method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [Appl. Numer. Math. 45 (2003) 411–418], and we prove it is convergent, and that...
We propose a Lagrangian approach to deriving energy-preserving finite difference schemes for the Euler–Lagrange partial differential equations. Noether’s theorem states that the symmetry of time translation of Lagrangians yields the energy conservation law. We introduce a unique viewpoint on this theorem: “the symmetry of time translation of Lagrangians derives the Euler–Lagrange equation and the energy conservation law, simultaneously.” The proposed method is a combination of a discrete counter...
In long-time numerical integration of Hamiltonian systems,
and especially in molecular dynamics simulation,
it is important that the energy is well conserved. For symplectic
integrators applied with sufficiently small step size, this
is guaranteed by the existence of a modified
Hamiltonian that is exactly conserved up to exponentially small
terms. This article is concerned with the simplified
Takahashi-Imada method, which is a modification
of the Störmer-Verlet method that is as easy to implement...
In this paper, we study
the linear Schrödinger equation over the d-dimensional torus,
with small values of the perturbing potential.
We consider numerical approximations of the associated solutions obtained
by a symplectic splitting method (to discretize the time variable) in combination with the
Fast Fourier Transform algorithm (to discretize the space variable).
In this fully discrete setting, we prove that the regularity of the initial
datum is preserved over long times, i.e. times that are...
We study the enclosing problem for discrete and continuous dynamical systems in the context of computer assisted proofs. We review and compare the existing methods and emphasize the importance of developing a suitable set arithmetic for efficient algorithms solving the enclosing problem.
The parareal in time algorithm allows for efficient parallel numerical simulations of time-dependent problems. It is based on a decomposition of the time interval into subintervals, and on a predictor-corrector strategy, where the propagations over each subinterval for the corrector stage are concurrently performed on the different processors that are available. In this article, we are concerned with the long time integration of Hamiltonian systems. Geometric, structure-preserving integrators are...
The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh,...
The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle
easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh,...
The main objective of this paper is to prove
new necessary conditions to the existence of
KAM tori.
To do so, we develop a
set of
explicit a-priori estimates for smooth
solutions of Hamilton-Jacobi equations,
using a combination of methods from
viscosity solutions,
KAM and Aubry-Mather theories.
These estimates
are valid
in any
space dimension, and can be checked numerically
to detect gaps between KAM tori and Aubry-Mather sets.
We apply these results to detect non-integrable regions in
several...
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