Error propagation in numerical approximations near relative equilibria

Authors:
J. Álvarez;A. Durán
Affiliations:
Departamento de Matemática Aplicada, E.T.S. de Ingenieros Industriales, Universidad de Valladolid, Paseo del Cauce s/n, E-47011 Valladolid, Spain;Departamento de Matemática Aplicada, E.T.S.I. de Telecomunicación, Universidad de Valladolid, Campus Miguel Delibes, Camino del Cementerio s/n E-47011 Valladolid, Spain
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

Citing 14
Cited 1

Invariants and numerical methods for ODEs

Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems

Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Unitary integrators and applications to continuous orthonormalization techniques

SIAM Journal on Numerical Analysis
Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I

Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
Error Growth in the Numerical Integration of Periodic Orbits, with Application to Hamiltonian and Reversible Systems

SIAM Journal on Numerical Analysis
Backward Error Analysis for Numerical Integrators

SIAM Journal on Numerical Analysis
Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations

Journal of Computational Physics
Geometric integrators for the nonlinear Schrödinger equation

Journal of Computational Physics
Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations

SIAM Journal on Numerical Analysis
Numerical behaviour of stable and unstable solitary waves

Applied Numerical Mathematics
Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

Mathematics of Computation
A technique to construct symmetric variable-stepsize linear multistep methods for second-order systems

Mathematics of Computation
Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations

Numerische Mathematik
Symmetric multistep methods over long times

Numerische Mathematik

On the influence of numerical preservation of invariants when simulating Hamiltonian relative periodic orbits

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	7.29

Visualization

Abstract

We study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of Hamiltonian differential equations with symmetries. First it is shown that taking an initial perturbation of a relative equilibrium, the corresponding solution is related, in a first approximation, to another relative equilibrium, with the parameters perturbed from the original. Then, this is used to prove that, for stable relative equilibria, error growth with respect to the perturbed solution is in general quadratic, but only linear for schemes that preserve the invariant quantities of the problem. In this sense, the conclusion is similar to the one obtained when integrating unperturbed relative equilibria. Numerical experiments illustrate the results.