## Geometric Integrators for ODEs (2006)

Venue: | J. Phys. A |

Citations: | 18 - 5 self |

### BibTeX

@ARTICLE{Mclachlan06geometricintegrators,

author = {Robert I Mclachlan and G Reinout and W Quispel},

title = {Geometric Integrators for ODEs},

journal = {J. Phys. A},

year = {2006},

volume = {39},

pages = {5251--5285}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within round-off error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature. Geometric Integrators for ODEs 2 1.

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Citation Context ... of ϕ such as symplecticity [89]. Symmetries are a partial exception; the composition ϕSϕS is not S-equivariant, but it is closer to equivariant than ϕ is, when S 2 = 1 [56]. 2.4. Variational methods =-=[73]-=- Many ODEs and PDEs of mathematical physics are derived from variational principles with natural discrete analogs. For an ODE with Lagrangian L(q, ˙q), one can construct an approximate discrete Lagran... |

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Citation Context ... and not symplectic, but appears to have excellent stability properties for large time steps on linear and nonlinear problems. The “mollified impulse method” of García-Archilla, Sanz-Serna, and Skeel =-=[37]-=- applies to problems of type (ii). Suppose the system is described by H = Hf(p, q)+Vs(q), where �H ′′ ′′ f � ≫ �V s �. The method is the composition e Nτ/2J ∇ ¯ Vs (ϕτ(Hf)) N e Nτ/2J ∇ ¯ Vs , (68) whe... |

44 |
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Citation Context ...smooth problems involving collisions [97]; and powerful “asynchronous” variational integrators can be constructed, which use different, even incommensurate time steps on different parts of the system =-=[68]-=-. In these situations variational integrators appear to be natural, and to work extremely well in practice, even if the reason for their good performance (e.g., by preserving some geometric feature) i... |

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Citation Context ...d the much cruder fourth-order method ϕ(zτ)ϕ((1 − 2z)τ)ϕ(zτ), where z = (2 − 21/3 ) −1 and ϕ is standard leapfrog (Eq. (1)). This method, the first fourth-order symplectic integrator to be discovered =-=[28, 110, 119]-=- has errors 100 to 1000 times larger than LF4. Figure 4 shows the dynamics of this small-amplitude orbit out to t = 20000. The periods of the orbit are about 2π and 28084. Remarkably, the integrator L... |

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Citation Context ...We also consider Hamiltonian systems with a dissipative perturbation. More details on these topics and extensive references can be found in several recent books [44, 67, 103, 109] and survey articles =-=[17, 18, 33, 45, 57, 69, 71, 83, 84]-=-. Physical applications are ubiquitous, varying from celestial mechanics [62], via particle accelerators [35], to molecular dynamics [65] and many areas in between (see, e.g., applications in quantum ... |

33 | A.: Geometric integration: numerical solution of differential equations on manifolds
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Citation Context ...We also consider Hamiltonian systems with a dissipative perturbation. More details on these topics and extensive references can be found in several recent books [44, 67, 103, 109] and survey articles =-=[17, 18, 33, 45, 57, 69, 71, 83, 84]-=-. Physical applications are ubiquitous, varying from celestial mechanics [62], via particle accelerators [35], to molecular dynamics [65] and many areas in between (see, e.g., applications in quantum ... |

32 |
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Citation Context ...d leapfrog until the orbit enters the difficult region, when it automatically switches to the (more expensive) implicit method. A related method has been proposed for the N-body gravitational problem =-=[24]-=-. The system has Hamiltonian H = H1 + H2 where H2 is singular at collisions. The leapfrog method is applied to the splitting H = (H1 + (1 − K)H2) + KH2 where the monitor function K is equal to 0 near ... |

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Citation Context ...e J (x). Splitting is the most practical method when it applies, the most famous example being the free rigid body, a Poisson, momentum-preserving integrator being just a sequence of planar rotations =-=[74, 101, 65]-=-. Methods that apply to arbitrary Hamiltonians for particular Poisson structures rely on finding either canonical coordinates or symplectic coordinates (a Poisson map from a symplectic vector space to... |

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Citation Context ... steps, typically using orthogonal projection with respect to a suitable metric. For example, energy-preserving integrators have been constructed using discrete gradient methods, a form of projection =-=[88, 92]-=-. Although it is still used, projection is something of a last resort, as it typically destroys other properties of the method (such as symplecticity) and may not give good long-time behaviour. Revers... |

28 |
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Citation Context ..., it is also possible to split the system into linear and nonlinear parts, solving the linear part exactly. This is the approach traditionally used for the nonlinear Schrödinger equation, for example =-=[116]-=-. This has the advantage of removing any stability restriction due to the splitting of the oscillators, and if the nonlinearity is small, we have the advantages discussed above for near-integrable sys... |

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Citation Context ...sible, as in the long-time integration of Hamiltonian systems; (iv) it may be essential to obtain a useful (e.g. convergent) method, as in the discrete differential complexes used in electromagnetism =-=[2, 52]-=-; and (v) the development of general-purpose methods may be nearing completion, as in Runge–Kutta methods for ODEs. Most geometric properties are not preserved by traditional numerical methods. The re... |

24 | On Magnus integrators for time-dependent Schrödinger equations
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Citation Context ...ar problems ˙x = γ(a(t), x), such as the timedependent Schrödinger equation. Again they allow one to use evaluations, commutators, and exponentials of a to approximate xk+1 in the correct group orbit =-=[12, 50, 55]-=-. Splitting methods are more ad-hoc. One approach is to choose a basis for g, say v1, . . . , vn, and write a(x) = � ai(x)vi. The vector field γ(ai(x), x) is tangent to the one-dimensional group orbit... |

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Citation Context ...alled Lie-Poisson case. Such J always arise from symplectic reduction of a canonical system on T ∗ G where G is a Lie group; a symplectic integrator for this system can reduce to a Poisson integrator =-=[91]-=-. 3.8. Preserving phase-space volume Because the preservation of phase-space volume plays an important role in many applications (e.g. in incompressible fluid flow), we here present volume-preserving ... |

22 |
Time-discretized variational formulation of non-smooth frictional contact
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Citation Context ... form it preserves; singular Lagrangians can be treated; it suggests a natural treatment of holonomic (position) and nonholonomic (velocity) constraints and of nonsmooth problems involving collisions =-=[97]-=-; and powerful “asynchronous” variational integrators can be constructed, which use different, even incommensurate time steps on different parts of the system [68]. In these situations variational int... |

21 | A variational complex for difference equations
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Citation Context ...sible, as in the long-time integration of Hamiltonian systems; (iv) it may be essential to obtain a useful (e.g. convergent) method, as in the discrete differential complexes used in electromagnetism =-=[2, 52]-=-; and (v) the development of general-purpose methods may be nearing completion, as in Runge–Kutta methods for ODEs. Most geometric properties are not preserved by traditional numerical methods. The re... |

19 |
An algebraic approach to invariant preserving integators: the case of quadratic and Hamiltonian invariants., in "Numer
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Citation Context ...sfies the multistep formula. (It always exists, at least as a formal power series in τ.) Often the dynamics of ϕ dominates the long-term behaviour of the multistep method. Recently it has been proved =-=[25, 43]-=- that the underlying one-step methods for a class of time-symmetric multistep methods for second-order problems ¨x = f(x) are conjugate to symplectic, which explains their near-conservation of energy ... |

18 | Integrators for nonholonomic mechanical systems
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Citation Context ...hich generates the nonholonomic equations of motion can be discretized to yield useful integrators which preserve the key geometric features of the equations, namely the constraints and reversibility =-=[27, 80]-=-. For example, if the Lagrangian 1 2� ˙q�2 2 − V (q) is subject to the nonholonomic constraint A(q) ˙q = 0, then one choice of discrete Lagrangian yields the nonholonomic analogue of RATTLE, namely qk... |

16 | Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods
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Citation Context ...ized leapfrog algorithm, Eq. (7) below (‘LF2’ in the figures). We compare this to a fourth-order (composition, i.e. leapfrog-like) method with 6 force evaluations per time step due to Blanes and Moan =-=[14]-=- that has been tuned for this type of splitting (see Eq. (56)) (‘LF4’), and to two non-geometric, standard algorithms: the ‘classic’ fourth-order Runge–Kutta method with fixed step size (‘RK4’), and a... |

16 |
A study of extrapolation methods based on multistep schemes without parasitic solutions
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Citation Context ...igenvalues of Ω. In this case the mollified impulse method reduces to qk+1 − 2 cos(τΩ)qk + qk−1 = −τ 2 ψ(τΩ)∇Vs(φ(τΩ)qk) (70) with ψ(ξ) = sin(ξ)/ξ and φ(ξ) = 1, a method first introduced by Deuflhard =-=[30]-=-. In fact, the method (70) is time-symmetric, second order, and exact when ∇Vs(q) = const. for any even functions ψ, φ with ψ(0) = φ(0) = 1. It is symplectic if ψ(ξ) = (sin(ξ)/ξ)φ(ξ). It suffers from ... |

15 | Numerical integrators that preserve symmetries and reversing symmetries
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Citation Context ...iven by s� Xi = xk + τ aijf(Xj), xk+1 = xk + τ j=1 s� bjf(Xj). j=1 RK methods are “linear”, that is, the map from vector field f to the map xk ↦→ xk+1 commutes with linear changes of variable x ↦→ Ax =-=[89]-=-. (Alternatively, the method is independent of the basis of R n ). This implies, for example, that all RK methods preserve (17)sGeometric Integrators for ODEs 12 all linear symmetries of the system. T... |

14 | 2001], A reversible averaging integrator for multiple time-scale dynamics
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Citation Context ...ach case the new methods allow the time step to be increased significantly, by a factor of 5 or more, while still obtaining reliable results. The “reversible averaging” method of Leimkuhler and Reich =-=[66]-=- applies to problems of type (i). The fast subsystem is still integrated with a small time step, the slow variables being interpolated as needed, while the slow subsystem is integrated with a large ti... |

14 | A review of exponential integrators for first order semi-linear problems
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Citation Context ...k+1 = xk + τ j=1 s� bjf(Xj) + τ 2 j=1 j,l=1 s� bjlf ′ (Xj)(f(Xl)) + . . . . Three special cases are (i) exponential integrators, in which an analytic function of the Jacobian f ′ (Xj) is incorporated =-=[93]-=-; (ii) elementary differential (EDRK) methods, in which all terms of each elementary differential are evaluated at a single Xi [95], and (iii) multiderivative (MDRK) methods, in which, in addition, on... |

14 |
Symplectic correctors”, in Integration Algorithms and Classical
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(Show Context)
Citation Context ...s have been considered in order to find more efficient methods. We consider the two that apply most generally here. The first is the use of a “corrector” (also known as processing or effective order) =-=[6, 9, 118]-=-. Suppose the method ϕ can be factored as (11) (12) ϕ = χψχ −1 . (13) Then to evaluate n time steps, we have ϕ n = χψ n χ −1 , so only the cost of ψ is relevant. The maps ϕ and ψ are conjugate by the ... |

13 | Geometric Integration and its application
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(Show Context)
Citation Context ...We also consider Hamiltonian systems with a dissipative perturbation. More details on these topics and extensive references can be found in several recent books [44, 67, 103, 109] and survey articles =-=[17, 18, 33, 45, 57, 69, 71, 83, 84]-=-. Physical applications are ubiquitous, varying from celestial mechanics [62], via particle accelerators [35], to molecular dynamics [65] and many areas in between (see, e.g., applications in quantum ... |

13 |
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(Show Context)
Citation Context ...d the much cruder fourth-order method ϕ(zτ)ϕ((1 − 2z)τ)ϕ(zτ), where z = (2 − 21/3 ) −1 and ϕ is standard leapfrog (Eq. (1)). This method, the first fourth-order symplectic integrator to be discovered =-=[28, 110, 119]-=- has errors 100 to 1000 times larger than LF4. Figure 4 shows the dynamics of this small-amplitude orbit out to t = 20000. The periods of the orbit are about 2π and 28084. Remarkably, the integrator L... |

12 | 1999], A time-reversible variable-stepsize integrator for constrained dynamics - Barth, Leimkuhler, et al. |

12 |
Symplectic integrators with processing: a general study
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(Show Context)
Citation Context ...s have been considered in order to find more efficient methods. We consider the two that apply most generally here. The first is the use of a “corrector” (also known as processing or effective order) =-=[6, 9, 118]-=-. Suppose the method ϕ can be factored as (11) (12) ϕ = χψχ −1 . (13) Then to evaluate n time steps, we have ϕ n = χψ n χ −1 , so only the cost of ψ is relevant. The maps ϕ and ψ are conjugate by the ... |

12 | High order symplectic integrators for perturbed Hamiltonian systems
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(Show Context)
Citation Context ...ach order O(ετ n ) and ⌊ 1 2 (n−1)⌋ of order O(ε2 τ n ). One can easily build, for example, a 2 stage method of order O(ε 2 τ 4 + ε 3 τ 3 ), a 3 stage method of order O(ε 2 τ 6 + ε 3 τ 4 ), and so on =-=[10, 63, 76]-=-. This idea combines particularly well with the use of correctors. For any composition, even standard leapfrog, for all n there is a corrector that eliminates the O(ετ p ) error terms for all 1 < p < ... |

11 |
Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems Nonlinearity 13
- Shang
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(Show Context)
Citation Context ...ometric properties (usually a necessary condition for preservation of KAM tori). Proofs that geometric integrators actually do preserve KAM tori seem to mainly have been given for Hamiltonian systems =-=[44, 94, 104]-=- and reversible systems [44]. The step size must not be resonant with the frequencies of the torus. Of course, the torus does undergo an O(τ p ) shift in its position. Preservation of periodic orbits ... |

11 |
Solving linear partial differential equations by exponential splitting
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(Show Context)
Citation Context ...498, b5 = c2 = −0.3667132690474257, b6 = c1 = 0.1303114101821663. The coefficients of τ, called variously ai, bi, ci, and di above, cannot all be positive if the order of the method is greater than 2 =-=[7, 105]-=-. This can prevent the application of this kind of method to dissipative systems. Various extensions have been considered in order to find more efficient methods. We consider the two that apply most g... |

10 |
Computation of essential molecular dynamics by subdivision techniques
- Schütte
- 1999
(Show Context)
Citation Context ...of the Markov chain yields approximates of global quantities like invariant measures and almost invariant sets. Although the methods can be successfully applied to low-dimensional Hamiltonian systems =-=[31]-=-, as far as we know the question of what geometric structure the Markov chain can inherit from the ODE has not yet been examined.sGeometric Integrators for ODEs 37 Poincaré maps The computation of Poi... |

10 | Explicit, time reversible, adaptive step size control
- Hairer, Soderlind
(Show Context)
Citation Context ...egligible. There is more scope for transforming reversible systems by dt d˜t = g(x). When the original system is Hamiltonian and separable, there are reversible integrators for the transformed system =-=[67, 47]-=- that are either explicit or semiexplicit (require solving just a scalar nonlinear equation). Now there is essentially no cost to including adaptivity, but symplecticity has been sacrificed, raising q... |

10 | Composition methods in the presence of small parameters
- McLachlan
- 1995
(Show Context)
Citation Context ... of four particularly useful fourth-order methods, namely Eqs. (8), (11), (15), and (56) below. For other methods, and especially for higher-order methods, we refer the reader to the original sources =-=[6, 9, 10, 11, 12, 13, 14, 22, 23, 26, 57, 70, 75, 76, 77, 95, 108]-=-. 2.2. Runge–Kutta-like methods Runge–Kutta (RK) methods are defined for systems with linear phase space R n [20]. For the system ˙x = f(x), x(0) = x0, x ∈ R n , (16) the s-stage RK method with parame... |