On manifolds of connecting orbits in discretizations of dynamical systems
Nonlinear Analysis: Theory, Methods & Applications
Multi-symplectic integration methods for Hamiltonian PDEs
Future Generation Computer Systems - Special issue: Geometric numerical algorithms
Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones
Mathematics of Computation
Mathematics of Computation
Are high order variable step equistage initializers better than standard starting algorithms?
Journal of Computational and Applied Mathematics
Shadow hybrid Monte Carlo: an efficient propagator in phase space of macromolecules
Journal of Computational Physics
Important Aspects of Geometric Numerical Integration
Journal of Scientific Computing
Computing statistics for Hamiltonian systems: A case study
Journal of Computational and Applied Mathematics
Stability of Asynchronous Variational Integrators
Proceedings of the 21st International Workshop on Principles of Advanced and Distributed Simulation
Time behaviour of the error when simulating finite-band periodic waves. The case of the KdV equation
Journal of Computational Physics
Some conservation issues for the dynamical cores of NWP and climate models
Journal of Computational Physics
Stability of asynchronous variational integrators
Journal of Computational Physics
Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations
Mathematics and Computers in Simulation
Progress in scaling biomolecular simulations to petaflop scale platforms
Euro-Par'06 Proceedings of the CoreGRID 2006, UNICORE Summit 2006, Petascale Computational Biology and Bioinformatics conference on Parallel processing
Error propagation in numerical approximations near relative equilibria
Journal of Computational and Applied Mathematics
Journal of Computational Physics
A Second-Order Strong Method for the Langevin Equations with Holonomic Constraints
SIAM Journal on Scientific Computing
Variational collision integrator for polymer chains
Journal of Computational Physics
On the estimation and correction of discretization error in molecular dynamics averages
Applied Numerical Mathematics
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Backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods. This is true in particular for the integration of Hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time. Such results are typically based on two aspects of backward error analysis: (i) It can be shown that the modified vector fields have some qualitative properties which they share with the given problem and (ii) an estimate is given for the difference between the best interpolating vector field and the numerical method. These aspects have been investigated recently, for example, by Benettin and Giorgilli in [ J. Statist. Phys., 74 (1994), pp. 1117--1143], by Hairer in [Ann. Numer. Math., 1 (1994), pp. 107--132], and by Hairer and Lubich in [ Numer. Math., 76 (1997), pp. 441--462]. In this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs. Our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit Taylor series expansion of the numerical method and the corresponding flow maps as in the above-cited works. As an application we discuss the long time integration of chaotic Hamiltonian systems and the approximation of time averages along numerically computed trajectories.