Variational collocation on finite intervals

Prolongation-collocation Variational Integrators

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 construction of the discrete Lagrangian adopts Hermite interpolation polynomials and the Euler–Maclaurin quadrature formula, and involves applying collocation...

Spectral-collocation variational integrators

Spectral methods are a popular choice for constructing numerical approximations for smooth problems, as they can achieve geometric rates of convergence and have a relatively small memory footprint. In this paper, we introduce a general framework to convert a spectral-collocation method into a shootingbased variational integrator for Hamiltonian systems. We also compare the proposed spectral-col...

Collocation Methods for Boundary Value Problems on 'Long' Intervals

This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an /1-stable symmetric collocation method. Probl...

Tutorial Orthogonal Collocation on Finite Elements

On finite order variational sequences

We discuss intrinsic aspects of Krupka’s approach to finite–order variational sequences. We give intrinsic isomorphisms of the quotient subsheaves of the short finite– order variational sequence with sheaves of forms on jet spaces of suitable order, obtaining a new finite–order (short exact) variational sequence which is made by sheaves of polynomial differential operators. Moreover, we present...