Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems

Blanes, Sergio; Moan, P. C.

doi:10.1016/j.apnum.2005.11.004

Cited by 80 publications

(113 citation statements)

References 21 publications

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“…Moreover, other studies suggest that polynomial interpolation for the exponential matrix, such as the Real Leja points Method [68], converge as fast as Krylov methods without the memory requirements to save all the vectors defining the Krylov subspace. A Chebyshev approximation for the exponential of a matrix [69,70] is included for completeness, and also higher order exponential integrators based on the Magnus expansion with a commutator free formulation [71,72,73] are tested against a classical second order exponential integrator.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

Pintor¹

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SummaryThe neutron diffusion equation describes the neutron population in a nuclear reactor core. This work deals with this model for nuclear reactors with hexagonal geometries. First, the stationary neutron diffusion equation is studied. This is a differential eigenvalue problem, called Lambda modes problem. To solve the Lambda modes problem, different methods have been compared in one-dimensional geometries, resulting as the best one the Spectral Element Method. The operators are then discretized using this scheme in two-and three-dimensional geometries, and the resulting algebraic eigenvalue problem is solved with the implicit restarted Arnoldi method.Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration neutron diffusion equation. Initially, a one step backard Euler method is used to discretize this equation in time. The transients to test the behaviour of the method are based on moving the control rods of the reactor, simulating an accident where a control rod is ejected and a scram is intialized to control the power evolution. An unphysical behaviour appears when a node is partially rodded, the rod cusping effect, which is corrected by weighting the cross sections with the flux of the previous time step. Good results are obtained for the power evolution and the rod cusping effect is corrected for different transients. To solve the algebraic systems arising in the backward method, a Krylov method is used, and different preconditioning strategies are tested. The first one consists of using the block structure obtained by the energy groups to solve the system by blocks, and different acceleration techniques for the block iterative scheme and a preconditioner using this block structure are proposed. Also, a spectral preconditioner, which makes use of the information in a Krylov subspace to preconditionate the next system is studied. Second and fourth order exponential methods are also proposed to integrate the time dependent neutron diffusion equation, where the exponential of the system matrix has to be multiplied by a vector. These schemes allow us to work without building explicitly the system matrix, and different methods are compared to calculate the product of the system matrix by a vector, such as a Krylov method, a Chebyshev approximation method, and a Leja Points method.Some situations arise in which a set of modes of a nuclear reactor core have to be updated, as in perturbative calculations or in the use of modal methods. Updating this set of modes can be very expensive when using the Arnoldi method, and for this reason several methods based on a Newton Iteration are proposed, such as the Modified Block Newton method, a One Sided Block Newton method and a Two Sided Block Newton method. As an alternative strategy, a reduced order model to update a set of modes based on the Proper Generalized Decomposition is also proposed. This method obtains an approximated solution as a sum of separable functions over the whole domain, reduc...

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Section: Numerical Resultsmentioning

confidence: 99%

“…Alternatively, approximations up to the same order can be obtained by a product of exponentials of linear combinations of the α i which avoid the presence of commutators [71]. For example, we can consider…”

Section: Commutator Free Magnus Expansionmentioning

confidence: 99%

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

Pintor¹

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“…corresponds to a symmetric second order method, and , = 1 − , corresponds to a fourth-order method [22,98]. Notice that the Lie operators, since being derivatives, are written in the reverse order than the maps, and this is very important to keep in mind for non-linear non-autonomous problems in order to apply the method correctly (see [24] for more details on the Magnus series expansion and Magnus integrators for nonautonomous non-linear differential equations).…”

Section: Splitting Methods For Non-autonomous Problemsmentioning

confidence: 99%

“…One can design the schemes up to same order which involve the product of exponentials of linear combination of the ( ) instead of nested commutators. These schemes avoid the presence of commutators and preserve the same qualitative properties of the system [18,22]. One can consider the compositions of the form…”

Section: Commutator-free Magnus Integratorsmentioning

confidence: 99%

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Splitting methods for autonomous and non-autonomous perturbed equations

Seydaoğlu¹

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SummaryThis thesis addresses the treatment of perturbed problems with splitting methods. After motivating these problems in Chapter 1, we give a thorough introduction in Chapter 2, which includes the objectives, several basic techniques and already existing methods.In Chapter 3, we consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods for several numerical examples and present some new improved schemes.In Chapter 4, we propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum = + of a sparse and efficiently exponentiable matrix with sparse exponential and a dense matrix which is of small norm in comparison with . The predominant algorithm is based on scaling the large matrix by a small number 2 − , which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbation matrix in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought.In Chapter 5, we consider the numerical integration of the perturbed Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this chapter we present new sixth-and eighth-order symplectic exponential integrators that are tailored to the Hill's equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. Several numerical examples illustrate the performance of the new methods.Conclusions and pointers to further research are detailed in Chapter 6. I ResumenEsta tesis aborda el tratamiento de problemas perturbados con métodos de escisión (splitting). Tras motivar el origen de este t...

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Performance of fourth and sixth‐order commutator‐free Magnus expansion integrators for Ehrenfest dynamics

Pueyo

Blanes

Castro

2020

Comp and Math Methods

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Hybrid quantum‐classical systems combine both classical and quantum degrees of freedom. Typically, in Chemistry, Molecular Physics, or Materials Science, the classical degrees of freedom describe atomic nuclei (or cations with frozen core electrons), whereas the quantum particles are the electrons. Although many possible hybrid dynamical models exist, the basic one is the so‐called Ehrenfest dynamics that results from the straightforward partial classical limit applied to the full quantum Schrödinger equation. Few numerical methods have been developed specifically for the integration of this type of systems. Here we present a preliminary study of the performance of a family of recently developed propagators: the (quasi) commutator‐free Magnus expansions. These methods, however, were initially designed for nonautonomous linear equations. We employ them for the nonlinear Ehrenfest system, by approximating the state value at each time step in the propagation, using an extrapolation from previous time steps.

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Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems

Cited by 80 publications

References 21 publications

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

Splitting methods for autonomous and non-autonomous perturbed equations

Performance of fourth and sixth‐order commutator‐free Magnus expansion integrators for Ehrenfest dynamics

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