Symmetric linear multistep methods

Some important early contributions of Germund Dahlquist are reviewed and their impact to recent developments in the numerical solution of ordinary differential equations is shown. This work is an elaboration of a talk presented in the Dahlquist session at the SciCADE05 conference in Nagoy

Global modified Hamiltonian for constrained symplectic integrators

Summary.: We prove that the numerical solution of partitioned Runge-Kutta methods applied to constrained Hamiltonian systems (e.g., the Rattle algorithm or the Lobatto IIIA-IIIB pair) is formally equal to the exact solution of a constrained Hamiltonian system with a globally defined modified Hamiltonian. This property is essential for a better understanding of their longtime behaviour. As an illustration, the equations of motion of an unsymmetric top are solved using a parameterization with Euler parameter

Important aspects of geometric numerical integration

At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the ‘non-stiff' situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (‘stiff') case this theory breaks down. Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present. This paper terminates with numerical experiments at space discretizations of the sine-Gordon equation, where a whole spectrum of frequencies is presen

Thermodynamics of osp(1|2) Integrable Spin Chain: Finite Size Correction

This note is a supplement to our previous papers: Mod. Phys. Lett. A14 (1999) 2427 (math-ph/9911010); Int. J. Mod. Phys. A15 (2000) 2329 (math-ph/9912014). The thermodynamic Bethe ansatz (TBA) equation for an integrable spin chain related to the Lie superalgebra osp(1|2) is analyzed. The central charge determined by low temperature asymptotics of the specific heat can be expressed by the Rogers dilogarithmic function, and identified to be 1. Solving the TBA equation numerically, we evaluate the several thermodynamic quantities. The excited state TBA equation is also discussed.Comment: 13 pages, 1 figure, to appear in J. Phys. Soc. Jpn. Vol. 70-

Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions

For trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with one or several constant high frequencies, near-conservation of the total and oscillatory energies are shown over time scales that cover arbitrary negative powers of the step size. This requires non-resonance conditions between the step size and the frequencies, but in contrast to previous results the results do not require any non-resonance conditions among the frequencies. The proof uses modulated Fourier expansions with appropriately modified frequencies.Comment: 26 page

Energy-diminishing integration of gradient systems

For gradient systems in Euclidean space or on a Riemannian manifold the energy decreases monotonically along solutions. Algebraically stable Runge-Kutta methods are shown to also reduce the energy in each step under a mild step-size restriction. In particular, Radau IIA methods can combine energy monotonicity and damping in stiff gradient systems. Discrete-gradient methods and averaged vector field collocation methods are unconditionally energy-diminishing, but cannot achieve damping for very stiff gradient systems. The methods are discussed when they are applied to gradient systems in local coordinates as well as for manifolds given by constraint

Linear energy-preserving integrators forPoisson systems

For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge-Kutta method with infinitely many stage

Symmetric multistep methods over long times

Summary.: For computations of planetary motions with special linear multistep methods an excellent long-time behaviour is reported in the literature, without a theoretical explanation. Neither the total energy nor the angular momentum exhibit secular error terms. In this paper we completely explain this behaviour by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution component

On the Energy Distribution in Fermi-Pasta-Ulam Lattices

This paper presents a rigorous study, for Fermi-Pasta-Ulam (FPU) chains with large particle numbers, of the formation of a packet of modes with geometrically decaying harmonic energies from an initially excited single low-frequency mode and the metastability of this packet over longer time scales. The analysis uses modulated Fourier expansions in time of solutions to the FPU system, and exploits the existence of almost-invariant energies in the modulation system. The results and techniques apply to the FPU α- and β-models as well as to higher-order nonlinearities. They are valid in the regime of scaling between particle number and total energy in which the FPU system can be viewed as a perturbation to a linear system, considered over time scales that go far beyond standard perturbation theory. Weak non-resonance estimates for the almost-resonant frequencies determine the time scales that can be covered by this analysi