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For the kinetic energy of 1d model finite systems the leading corrections to local approximations as a functional of the potential are derived using semiclassical methods. The corrections are simple, non-local functionals of the potential.…
In this paper, we develop variational integrators for the nonequilibrium thermodynamics of simple closed systems. These integrators are obtained by a discretization of the Lagrangian variational formulation of nonequilibrium thermodynamics…
We present a strategy for solving time-dependent problems on grids with local refinements in time using different time steps in different regions of space. We discuss and analyze two conservative approximations based on finite volume with…
We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic, momentum preserving, and can be constructed to be of arbitrarily high-order, or can be made to converge…
We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent…
We propose a framework for the connection between local symmetries of discrete Hamiltonians and the design of compact localized states. Such compact localized states are used for the creation of tunable, local symmetry-induced bound states…
We present a discrete total variation calculus in Hamiltonian formalism in this paper. Using this discrete variation calculus and generating functions for the flows of Hamiltonian systems, we derive two-step symplectic-energy integrators of…
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 show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order system of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and…
A new exponentially fitted version of the Discrete Variational Derivative method for the efficient solution of oscillatory complex Hamiltonian Partial Differential Equations is proposed. When applied to the nonlinear Schroedinger equation,…
We investigate discretization strategies for a recently introduced class of energy-based models. The model class encompasses classical port-Hamiltonian systems, generalized gradient flows, and certain systems with algebraic constraints. Our…
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…
We derive a canonical form for smooth vector fields on $\Re^{n+1}$. We use this to demonstrate the local multi-Hamiltonian nature of the corresponding flows. Associated with the canonical form is an inhomogenious linear PDE whose solutions…
In this paper, we present a linearly implicit energy-preserving scheme for the Camassa-Holm equation by using the multiple scalar auxiliary variables approach, which is first developed to construct efficient and robust energy stable schemes…
We consider the problem of topological linearization of smooth (C infinity or real analytic) control systems, i.e. of their local equivalence to a linear controllable system via point-wise transformations on the state and the control…
We develop a semi-discrete version of discrete variational mechanics with applications to numerical integration of classical field theories. The geometric preservation properties are studied.
In this paper we consider discrete gradient methods for approximating the solution and preserving a first integral (also called a constant of motion) of autonomous ordinary differential equations. We prove under mild conditions for a large…
This paper discusses the construction of local bounded commuting projections for discrete subcomplexes of the gradgrad complexes in two and three dimensions, which play an important role in the finite element theory of elasticity (2D) and…
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that (linear) projection methods are a subset of discrete…
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are…