Geometric integrators for ODEs

McLachlan, Robert I.; Quispel, G.

doi:10.1088/0305-4470/39/19/s01

Cited by 156 publications

(165 citation statements)

References 112 publications

(162 reference statements)

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“…VI], [3] and references therein). Another challenging numerical task in conservative Hamiltonian systems is to discriminate between order and chaos.…”

Section: Introductionmentioning

confidence: 99%

Numerical integration of variational equations

Skokos

Gerlach

2010

Phys. Rev. E

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We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the 'tangent map (TM) method', a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic map S, while the corresponding tangent map T S, is used for the integration of the variational equations. A simple and systematic technique to construct T S is also presented.

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“…VI], [3] and references therein). Another challenging numerical task in conservative Hamiltonian systems is to discriminate between order and chaos.…”

Section: Introductionmentioning

confidence: 99%

Numerical integration of variational equations

Skokos

Gerlach

2010

Phys. Rev. E

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“…In this sense, schemes (1.3) can be considered as geometric integrators, and as such, they show smaller error growth than standard integrators. It is not surprising, then, that a systematic search for splitting methods of higher order of accuracy has taken place during the last two decades and a large number of them exist in the literature (see [12,17,25,26,29] and references therein) which have been specifically designed for different families of problems.…”

Section: Introductionmentioning

confidence: 99%

On the Linear Stability of Splitting Methods

Blanes

Casas

Murua³

2007

Found Comput Math

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A comprehensive linear stability analysis of splitting methods is carried out by means of a 2 × 2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible timereversible splitting schemes for a prescribed stability polynomial. It is shown that p(x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By selecting conveniently the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available.

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“…R. McLachlan (private communication) has noticed this for the Henon-Heiles (HeHe) problem [27], where a phase plane plot that is very different from the correct one is obtained. Here we concentrate on another instance.…”

Section: Integrating Hamiltonian Systems Using Ode45mentioning

confidence: 96%

Surprising computations

Ascher

2012

Applied Numerical Mathematics

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In the course of simulation of differential equations, especially of marginally stable differential problems using marginally stable numerical methods, one occasionally comes across a correct computation that yields surprising, or unexpected results. We examine several instances of such computations. These include (i) solving Hamiltonian ODE systems using almost conservative explicit Runge-Kutta methods, (ii) applying splitting methods for the nonlinear Schrödinger equation, and (iii) applying strong stability preserving Runge-Kutta methods in conjunction with weighted essentially non-oscillatory semi-discretizations for nonlinear conservation laws with discontinuous solutions.For each problem and method class we present a simple numerical example that yields results that in our experience many active researchers are finding unexpected and unintuitive. Each numerical example is then followed by an explanation and a resolution of the practical problem.

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Geometric integrators for ODEs

Cited by 156 publications

References 112 publications

Numerical integration of variational equations

Numerical integration of variational equations

On the Linear Stability of Splitting Methods

Surprising computations

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