Splitting methods with complex times for parabolic equations

Castella, François; Chartier, Philippe; Descombes, Stéphane; Vilmart, Gilles

doi:10.1007/s10543-009-0235-y

Cited by 82 publications

(147 citation statements)

References 21 publications

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“…Notice that multi-revolution composition methods with complex coefficients can also be considered (see [6,17] in the context of standard composition methods). For instance, the fourth order conditions to achieve order 3 for a multi-revolution composition method have a complex solution for s = 2, given for all N ≥ 2 by:…”

Section: Effective Construction Of Mrcmsmentioning

confidence: 99%

Multi-revolution composition methods for highly oscillatory differential equations

et al. 2014

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We introduce a new class of multi-revolution composition methods (MRCM) for the approximation of the N th -iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, the technique benefits from the properties of standard composition methods: it is intrinsically geometric and well-suited for Hamiltonian or divergence-free equations for instance. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schrödinger equation, illustrate the theoretical results, as well as the efficiency and versatility of the methods.

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Section: Effective Construction Of Mrcmsmentioning

confidence: 99%

Multi-revolution composition methods for highly oscillatory differential equations

et al. 2014

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“…Thalhammer [28], and Hansen and Ostermann [14], proved that splitting methods keep their classical order for certain classes of problems with unbounded operators, given that the solution is sufficiently regular. See also [5,15], which enables splitting methods of order higher than two for parabolic problems using the theory of analytic semigroups. Using the calculus of Lie derivatives, the theory was extended to non-linear problems in [17,29].…”

Section: Introductionmentioning

confidence: 99%

Stiff convergence of force-gradient operator splitting methods

Kieri

2015

Applied Numerical Mathematics

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Stiff convergence of force-gradient operator splitting methods. Applied Numerical Mathematics AbstractWe consider force-gradient, also called modified potential, operator splitting methods for problems with unbounded operators. We prove that force-gradient operator splitting schemes retain their classical orders of accuracy for linear time-dependent partial differential equations of parabolic and Schrödinger types, provided that the solution is sufficiently regular.

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“…This class of methods has been recently used for the numerical integration of the autonomous case, showing good performances [14,34,49].…”

Section: The Separable Non-autonomous Parabolic Equationsmentioning

confidence: 99%

“…In order to circumvent this order-barrier, the papers [34] and [49] simultaneously presented a systematic analysis for a class of composition methods with complex coefficients having positive real parts. Using this extension from the real line to the complex plane, the authors of [34] and [49] built up methods of orders 3 to 14 by considering a technique known as triple-jump composition.…”

Section: The Problemmentioning

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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Splitting methods with complex times for parabolic equations

Cited by 82 publications

References 21 publications

Multi-revolution composition methods for highly oscillatory differential equations

Multi-revolution composition methods for highly oscillatory differential equations

Stiff convergence of force-gradient operator splitting methods

Splitting methods for autonomous and non-autonomous perturbed equations

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