Related papers: Mid-point embedding of Hamiltonian systems and var…
We define discrete Hamiltonian systems in the framework of discrete embeddings. An explicit comparison with previous attempts is given. We then solve the discrete Helmholtz's inverse problem for the discrete calculus of variation in the…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
Numerical algorithms based on variational and symplectic integrators exhibit special features that make them promising candidates for application to general relativity and other constrained Hamiltonian systems. This paper lays part of the…
The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified…
We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The…
We present a new class of exponential integrators for ordinary differential equations. They are locally exact, i.e., they preserve the linearization of the original system at every point. Their construction consists in modifying existing…
Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In…
Discrete Hamiltonian variational integrators are derived from Type II and Type III generating functions for symplectic maps, and in this paper we establish a variational error analysis result that relates the order of accuracy of the…
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
This contribution presents an integration method based on the Simpson quadrature. The integrator is designed for finite-dimensional nonlinear mechanical systems that derive from variational principles. The action is discretized using…
Finite-dimensional non-canonical Hamiltonian systems arise naturally from Hamilton's principle in phase space. We present a method for deriving variational integrators that can be applied to perturbed non-canonical Hamiltonian systems on…
We construct several variational integrators--integrators based on a discrete variational principle--for systems with Lagrangians of the form L = L_A + epsilon L_B, with epsilon << 1, where L_A describes an integrable system. These…
The difference and similarity between the velocity- and position-Verlet integrators are discussed from the viewpoint of their Hamiltonian representations for both linear and nonlinear systems. For a harmonic oscillator, the exact…
We present a brief tutorial on the nuts and bolts computation of a multisymplectic particle-in-cell algorithm using the discretized Lagrangian approach. This approach, originated by Marsden, Shadwick, and others, brings the benefits of…
We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our…
The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive…
Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. Variational integrators are an important class of geometric integrators. The general idea…
The general topic of the present paper is to study the conservation for some structural property of a given problem when discretising this problem. Precisely we are interested with Lagrangian or Hamiltonian structures and thus with…
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this…
A new method is proposed for integrating the equations of motion of an elastic filament. In the standard finite-difference and finite-element formulations the continuum equations of motion are discretized in space and time, but it is then…