Related papers: Numerical comparisons among some methods for Hamil…
We present a comparison of different numerical techniques for the integration of variational equations. The methods presented can be applied to any autonomous Hamiltonian system whose kinetic energy is quadratic in the generalized momenta,…
Symplectic integration algorithms have become popular in recent years in long-term orbital integrations because these algorithms enforce certain conservation laws that are intrinsic to Hamiltonian systems. For problems with large variations…
We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D.Ebin's long-time existence result for geodesics on…
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system…
In a recent series of papers, the class of energy-conserving Runge-Kutta methods named Hamiltonian BVMs (HBVMs) has been defined and studied. Such methods have been further generalized for the efficient solution of general conservative…
On this paper, we have proposed an approach to observe the time-centered difference scheme for dissipative mechanical systems from a Hamiltonian perspective and to introduce the idea of symplectic algorithm to dissipative systems. The…
We discuss the use of methods coming from integrable systems to study problems of enumerative and algebraic combinatorics, and develop two examples: the enumeration of Alternating Sign Matrices and related combinatorial objects, and the…
Symplectic integrators constructed from Hamiltonian and Lie formalisms are obtained as symplectic maps whose flow follows the exact solution of a "sourrounded" Hamiltonian K = H + h^k H_1. Those modified Hamiltonians depends virtually on…
We detect, by using symplectic topology, invariant measures with large rotation vectors for a class of Hamiltonian flows.
Symplectic integrators are widely used for the study of planetary dynamics and other $N$-body problems. In a study of the outer Solar system, we demonstrate that individual symplectic integrations can yield biased errors in the semi-major…
In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall…
We consider the numerical integration of the matrix Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian…
Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to \emph{dissipative} systems is inherently difficult due to dissipative systems' lack…
Given a fluid equation with reduced Lagrangian $l$ which is a functional of velocity $\MM{u}$ and advected density $D$ given in Eulerian coordinates, we give a general method for semidiscretising the equations to give a canonical…
Efficient fourth order symplectic integrators are proposed for numerical integration of separable Hamiltonian systems H(p,q)=T(p)+V(q). Symmetric splitting coefficients with five to nine stages are obtained by higher order decomposition of…
This paper focuses on the numerical approximation of the linearized shallow water equations using hybridizable discontinuous Galerkin (HDG) methods, leveraging the Hamiltonian structure of the evolution system. First, we propose an…
In this paper we study arbitrarily high-order energy-conserving methods for simulating the dynamics of a charged particle. They are derived and studied within the framework of Line Integral Methods (LIMs), previously used for defining…
A fixed time-step variational integrator cannot preserve momentum, energy, and symplectic form simultaneously for nonintegrable systems. This barrier can be overcome by treating time as a discrete dynamic variable and deriving adaptive…
A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic…
The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best…