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21
Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 96 (18 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
On Magnus Integrators for TimeDependent Schrödinger Equations
 SIAM J. Numer. Anal
, 2002
"... Numerical methods based on the Magnus expansion are an ecient class of integrators for Schrodinger equations with timedependent Hamiltonian. Though their derivation assumes an unreasonably small time step size as would be required for a standard explicit integrator, the methods perform well even f ..."
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Cited by 24 (2 self)
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Numerical methods based on the Magnus expansion are an ecient class of integrators for Schrodinger equations with timedependent Hamiltonian. Though their derivation assumes an unreasonably small time step size as would be required for a standard explicit integrator, the methods perform well even for much larger step sizes. This favorable behavior is explained, and optimalorder error bounds are derived which require no or only mild restrictions of the step size. In contrast to standard integrators, the error does not depend on higher time derivatives of the solution, which is in general highly oscillatory.
On the Global Error of Discretization Methods for HighlyOscillatory Ordinary Differential Equations
, 2000
"... Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the globalerror ..."
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Cited by 21 (5 self)
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Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the globalerror envelope for RungeKutta and Magnus methods. Our results are closely matched by numerical experiments. Motivated by the superior performance of Liegroup methods, we present a modification of the Magnus expansion which displays even better longterm behaviour in the presence of oscillations.
High order optimized geometric integrators for linear differential equations
, 2000
"... In this paper new integration algorithms for linear differential equations up to eighth order are obtained. Starting from Magnus expansion, methods based on Cayley transformation and Fer expansion are also built. The structure of the exact solution is retained while the computational cost is reduced ..."
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Cited by 7 (1 self)
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In this paper new integration algorithms for linear differential equations up to eighth order are obtained. Starting from Magnus expansion, methods based on Cayley transformation and Fer expansion are also built. The structure of the exact solution is retained while the computational cost is reduced compared to similar methods. Their relative performance is tested on some illustrative examples.
Complexity theory for Liegroup solvers
, 1999
"... Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and M ..."
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Cited by 5 (0 self)
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Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and Magnus expansions, whose conclusion is that for order four, six and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Liegroup method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g. Krylovsubspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods.
Global error control of the timepropagation for the Schrödinger equation with an explicitly timedependent Hamiltonian
, 2009
"... We use a posteriori error estimation theory to derive a relation between local and global error in the propagation for the timedependent Schrödinger equation. Based on this result, we design a class of h, padaptive Magnus–Lanczos propagators capable of controlling the global error of the timestep ..."
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Cited by 4 (2 self)
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We use a posteriori error estimation theory to derive a relation between local and global error in the propagation for the timedependent Schrödinger equation. Based on this result, we design a class of h, padaptive Magnus–Lanczos propagators capable of controlling the global error of the timestepping scheme by only solving the equation once. We provide results for models of several different small molecules including bounded and dissociative states, illustrating the efficiency and wide applicability of the new methods. Key words global error control·h, padaptivity · Magnus–Lanczos propagator · timedependent Schrödinger equation 1
A Higher Order Local Linearization Method for Solving Ordinary Differential Equations
"... The Local Linearization (LL) method for the integration of ordinary differential equations is an explicit onestep method that has a number of suitable dynamical properties. However, a major drawback of the LL integrator is that its order of convergence is only two. The present paper overcomes this ..."
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Cited by 3 (0 self)
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The Local Linearization (LL) method for the integration of ordinary differential equations is an explicit onestep method that has a number of suitable dynamical properties. However, a major drawback of the LL integrator is that its order of convergence is only two. The present paper overcomes this limitation by introducing a new class of numerical integrators, called the LLT method, that is based on the addition of a correction term to the LL approximation. In this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the LL method. In particular, it is proved that the LLT method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence. The performance of the introduced method is further illustrated through computer simulations.
Sufficient conditions for the convergence of the Magnus expansion
, 2007
"... Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation Y′= A(t)Y. The first one provides a bound on the convergence domain based on the norm of the operator A(t). The second condition links the convergence of ..."
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Cited by 2 (0 self)
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Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation Y′= A(t)Y. The first one provides a bound on the convergence domain based on the norm of the operator A(t). The second condition links the convergence of the expansion with the structure of the spectrum of Y (t), thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.