On the necessity of negative coefficients for operator splitting schemes of order higher than two

Blanes, Sergio; Casas, Fernando

doi:10.1016/j.apnum.2004.10.005

Cited by 77 publications

(116 citation statements)

References 17 publications

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“…This exact form for g obviated the need to determine g's upper bound as it is done originally in the work of Suzuki 12 , and in the more recent work on symplectic correctors 7 .) With λ 1 and λ 2 known, the minimium of F is given by 24) and therefore,…”

Section: A Constructive Proof Of the Sheng-suzuki Theoremmentioning

confidence: 99%

Structure of positive decompositions of exponential operators

Chin

2005

Phys. Rev. E

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The solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data. Accurate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schrödinger equation, it is immaterial whether or not the decomposition coefficients are positive. In fact, most symplectic algorithms for solving classical dynamics contain some negative coefficients. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions, which respect the time-irreversibility of the diffusion kernel, can yield practical algorithms. These positive time steps only, forward decompositions, are a highly effective class of factorization algorithms. This work introduce a framework for understanding the structure of these algorithms. By a suitable representation of the factorization coefficients, we show that specific error terms and order conditions can be solved analytically. Using this framework, we can go beyond the Sheng-Suzuki theorem and derive a lower bound for the error coefficient eV T V . By generalizing the framework perturbatively, we can further prove that it is not possible to have a sixth order forward algorithm by including only theThe pattern of these higher order forward algorithms is that in going from the (2n) th to the (2n+2) th order, one must include a new commutator [V T 2n−1 V ] in the decomposition process.

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Section: A Constructive Proof Of the Sheng-suzuki Theoremmentioning

confidence: 99%

Structure of positive decompositions of exponential operators

Chin

2005

Phys. Rev. E

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“…The 'exact' solution is computed with a very small time step. We observe the expected orders (lines of slopes 2,4,6,8). Surprisingly, composition methods using the PeacemanRachford formula are slightly more accurate than the one using exponentials.…”

Section: The Linear Casementioning

confidence: 88%

“…The existence of at least one negative coefficient was shown in [20,21], and the existence of a negative coefficient for both operators was proved in [12]. An elegant geometric proof can also be found in [2]. As a consequence, such high-order splitting methods cannot be used in general when one operator A or B has large negative spectrum, or when it only generates a C 0 semi-group of propagators -and not a group (like the Laplacian).…”

Section: Introductionmentioning

confidence: 99%

Splitting methods with complex times for parabolic equations

et al. 2009

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Using composition procedures, we build up high order splitting methods to solve evolution equations posed in finite or infinite dimensional spaces. Since high-order splitting methods with real time are known to involve large and/or negative time steps, which destabilizes the overall procedure, the key point of our analysis is, we develop splitting methods that use complex time steps having positive real part: going to the complex plane allows to considerably increase the accuracy, while keeping small time steps; on the other hand, restricting our attention to time steps with positive real part makes our methods more stable, and in particular well adapted in the case when the considered evolution equation involves unbounded operators in infinite dimensional spaces, like parabolic (diffusion) equations.We provide a thorough analysis in the case of linear equations posed in general Banach spaces. We also numerically investigate the nonlinear situation. We illustrate our results in the case of (linear and nonlinear) parabolic equations.

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“…We extend the results of Bayer and Teichmann [2], where strong conditions are imposed on the vector fields, to more general coefficients and test functions. This allows us to obtain methods of order higher than 2 without having to resort to extrapolation, see Blanes and Casas [3] and Oshima et al [27]. The weighted spaces developed originally in Röckner and Sobol [29] and used for the numerical analysis of weak approximation methods in Dörsek and Teichmann [11] are a suitable tool for our needs, and we provide a refined analysis of the vector fields defined on these spaces.…”

Section: Introductionmentioning

confidence: 99%

Cubature methods for stochastic (partial) differential equations in weighted spaces

Dörsek

Teichmann

Velušček

2013

Stoch PDE: Anal Comp

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The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded test functions. We first describe a recently introduced flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da PratoZabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced technical assumption on weak symmetry of the cubature formula. Computational results for a cubature discretization of a spatially extended stochastic FitzHugh-Nagumo model, an SPDE model from mathematical biology, are shown, illustrating the applicability of our theory.

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On the necessity of negative coefficients for operator splitting schemes of order higher than two

Cited by 77 publications

References 17 publications

Structure of positive decompositions of exponential operators

Structure of positive decompositions of exponential operators

Splitting methods with complex times for parabolic equations

Cubature methods for stochastic (partial) differential equations in weighted spaces

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