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18
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
Variational integrators for constrained dynamical systems
, 2007
"... Key words Variational time integration, constrained dynamical systems, differential algebraic equations, flexible multibody dynamics. A variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constr ..."
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Key words Variational time integration, constrained dynamical systems, differential algebraic equations, flexible multibody dynamics. A variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constrained problems is extended by aspects on the initialization of simulations, the discrete Legendre transform and certain postprocessing steps. Furthermore, the discrete null space method which has been introduced in the framework of energymomentum conserving integration of constrained systems is adapted to the framework of variational integrators. It eliminates the constraint forces (including the Lagrange multipliers) from the timestepping scheme and subsequently reduces its dimension to the minimal possible number. While retaining the structure preserving properties of the specific integrator, the solution of the smaller dimensional system saves computational costs and does not suffer from conditioning problems. The performance of the variational discrete null space method is illustrated by numerical examples dealing with mass point systems, a closed kinematic chain of rigid bodies and flexible multibody dynamics and the solutions are compared to those obtained by an energymomentum scheme. © 2008 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim 1
Splitting and composition methods in the numerical integration of differential equations
, 2008
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On multisymplecticity of partitioned Runge–Kutta and splitting methods
"... Although Runge–Kutta and partitioned Runge–Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multiHamiltonian PDEs, they do not always lead to welldefined numerical methods. WeconsiderthecasestudyofthenonlinearSchrödinger equation in detail, for ..."
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Although Runge–Kutta and partitioned Runge–Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multiHamiltonian PDEs, they do not always lead to welldefined numerical methods. WeconsiderthecasestudyofthenonlinearSchrödinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on the (second order) box scheme, and construct welldefined, explicit integrators, of various orders, with local discrete multisymplectic conservation laws, based on partitioned Runge–Kutta methods. We also show that two popular explicit splitting methods are multisymplectic.
The KelvinHelmholtz instability of momentum sheets in the Euler equations for planar diffeomorphisms
 SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
, 2006
"... The Euler equations that describe geodesics on the group of diffeomorphisms of the plane admit singular solutions in which the momentum is concentrated on curves, the socalled momentum sheets analogous to vortex sheets in the Euler fluid equations. We study the stability of straight and circular ..."
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The Euler equations that describe geodesics on the group of diffeomorphisms of the plane admit singular solutions in which the momentum is concentrated on curves, the socalled momentum sheets analogous to vortex sheets in the Euler fluid equations. We study the stability of straight and circular momentum sheets for a large family of metrics. We prove that straight sheets moving normally to themselves under an H 1 metric, corresponding to peakons for the onedimensional (1D) Camassa–Holm equation, are linearly stable in Eulerian coordinates, suffering only a weak instability of Lagrangian particle paths, while most other cases are unstable but wellposed. Expanding circular sheets are algebraically unstable for all metrics. The evolution of the instabilities are followed numerically, illustrating several typical dynamical phenomena of momentum sheets.
A NEW IMPLEMENTATION OF SYMPLECTIC RUNGE–KUTTA METHODS ∗
, 1637
"... Abstract. We propose a “Newton–Taylor ” iteration for solving the implicit equations of symplectic Runge–Kutta methods, using the Jacobian of the vector field and matrixvector multiplications whose extra cost for certain structured problems is negligible. The structure of Hamiltonian ODEs allows th ..."
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Abstract. We propose a “Newton–Taylor ” iteration for solving the implicit equations of symplectic Runge–Kutta methods, using the Jacobian of the vector field and matrixvector multiplications whose extra cost for certain structured problems is negligible. The structure of Hamiltonian ODEs allows this very simple iteration to be effective. The iteration reduces the number of vector field evaluations almost to that of Newton’s method, often only one or two per time step, making symplectic Runge–Kutta methods more efficient even at relatively large time steps. Key words. Runge–Kutta, implicit, symplectic integrators, inexact Newton methods AMS subject classifications. 65L06, 65P10
LAGRANGED’ALEMBERT SPARK INTEGRATORS FOR NONHOLONOMIC LAGRANGIAN SYSTEMS
"... Abstract. We consider Lagrangian systems with ideal nonholonomic constraints. These systems can be expressed as implicit index 2 differentialalgebraic equations (DAEs) and can be derived from the Lagranged’Alembert principle. Methods based on a discrete Lagranged’Alembert principle are called Lag ..."
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Abstract. We consider Lagrangian systems with ideal nonholonomic constraints. These systems can be expressed as implicit index 2 differentialalgebraic equations (DAEs) and can be derived from the Lagranged’Alembert principle. Methods based on a discrete Lagranged’Alembert principle are called Lagranged’Alembert integrators and they generalize variational integrators. We define a new nonholonomically constrained discrete Lagranged’Alembert principle based on a discrete Lagranged’Alembert principle for forced Lagrangian systems. The principle that we propose does not make explicit use of any Lagrange multiplier in its formulation. Nonholonomic constraints are considered as first integrals of the underlying forced Lagrangian system of ordinary differential equations. We show that a large class of specialized partitioned additive RungeKutta (SPARK) methods for index 2 DAEs satisfies the new discrete principle. Symmetric Lagranged’Alembert SPARK integrators of any order can be obtained based for example on Gauss and Lobatto coefficients as already proposed for more general index 2 DAEs. Our results are illustrated by several numerical experiments. Key words. Differentialalgebraic equations, discrete mechanics, forcing, Gauss coefficients,
The applicability of constrained symplectic integrators in general relativity
 J. Phys. A: Math. Theor
, 2008
"... ABSTRACT. The purpose of this note is to point out that a naive application of symplectic integration schemes for Hamiltonian systems with constraints such as SHAKE or RATTLE which preserve holonomic constraints encounters difficulties when applied to the numerical treatment of the equations of gene ..."
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ABSTRACT. The purpose of this note is to point out that a naive application of symplectic integration schemes for Hamiltonian systems with constraints such as SHAKE or RATTLE which preserve holonomic constraints encounters difficulties when applied to the numerical treatment of the equations of general relativity. It is well known that the equations of General Relativity (GR) can be derived from a variational principle and that they can be cast into Hamiltonian form. The underlying symplectic structure has been studied as early as the 1940’s beginning with the work of Bergmann [3,8], Dirac [9,10] and ADM [5]. The main motivation then has been to work out a quantisation scheme for GR. For various reasons, not the least of them being the peculiar nature of the symplectic structure of GR, these early attempts have not led to any viable theory of quantum gravity. On the other hand it has been well established within the numerical mathematics community [12, 13, 16] that the use of so called symplectic integrators i.e., numerical ODE solvers which preserve an underlying symplectic structure can lead to significant improvements in longtime stability, conservation of first integrals
Communicated by Peter Olver.
, 2007
"... Abstract We present new explicit volumepreserving methods based on splitting for polynomial divergencefree vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the offdiagonal part and methods that do not. For the methods in the first cl ..."
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Abstract We present new explicit volumepreserving methods based on splitting for polynomial divergencefree vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the offdiagonal part and methods that do not. For the methods in the first class it is possible to combine different treatments of the diagonal and offdiagonal parts, giving rise to a number of possible combinations.