A Borel Transform Method for Locating Singularities of Taylor and Fourier Series

Pauls, Walter; Frisch, U.

doi:10.1007/s10955-007-9307-z

Cited by 25 publications

(37 citation statements)

References 41 publications

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“…Such a scheme can in principle be carrried over to the case with a boundary; this will, of course, require the numerical solution of the elliptic (Poisson) equations with both Dirichlet and Neumann boundary conditions involved in the Helmholtz-Hodge decompositions. For the case of a boundary that is not analytic, but in a suitable ultradifferential class, the time-Taylor series will generally be divergent for any t > 0, so that some resummation method (e.g., by a Borel transformation [55]) must also be used for the numerical implementation.…”

Section: Discussionmentioning

confidence: 99%

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

Besse¹,

Frisch²

2017

Commun. Math. Phys.

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The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries.For a good understanding, it is crucial to carry out, besides mathematical studies, highaccuracy and well-resolved numerical exploration. Such studies can be very demanding in computational resources, but recently it has been shown that very substantial gains can be achieved first, by using Cauchy's Lagrangian formulation of the Euler equations and second, by taking advantages of analyticity results of the Lagrangian trajectories for flows whose initial vorticity is Hölder-continuous. The latter has been known for about twenty years (Serfati, J. Math. Pures Appl. 74: 95-104, 1995), but the combination of the two, which makes use of recursion relations among time-Taylor coefficients to obtain constructively the time-Taylor series of the Lagrangian map, has been achieved only recently (Frisch and Zheligovsky, Commun. Math. Phys. 326: 499-505, 2014; Podvigina et al., J. Comput. Phys. 306: 320-342, 2016 and references therein).Here we extend this methodology to incompressible Euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity. Non-constructive regularity results for these cases have already been obtained by Glass et al. (Ann. Scient.Éc. Norm. Sup. 4 e série 45: 1-51, 2012). Using the invariance of the boundary under the Lagrangian flow, we establish novel recursion relations that include contributions from the boundary. This leads to a constructive proof of time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy-Lagrangian method.

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Section: Discussionmentioning

confidence: 99%

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

Besse¹,

Frisch²

2017

Commun. Math. Phys.

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“…We found that, when using 50-80 quadruple-precision Taylor coefficients from our L 2 series, such methods allow a determination of the radius of convergence with an error in the range 10%-20%. We also tried more advanced extrapolation techniques, such as convergence acceleration methods (see, e.g., [35,34] and references therein) and the asymptotic extrapolation technique [25,35,34], which have the potential of capturing both leading and subleading asymptotic behaviour. Unfortunately, all these techniques failed to reach the asymptotic regimes and thus gave no improvement over the more standard ones.…”

Section: Appendix B Computation Of the Radius Of Convergence Of A Pomentioning

confidence: 99%

The Cauchy–Lagrangian method for numerical analysis of Euler flow

Podvigina

Zheligovsky

Frisch

2016

Journal of Computational Physics

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A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by a recurrence relation that follows from the Cauchy invariants formulation of the Euler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution. Tests performed on the two-dimensional Euler equation indicate that the Cauchy-Lagrangian method is more -and occasionally much more -efficient and less prone to instability than Eulerian Runge-Kutta methods, and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algorithm. We also develop tools of analysis adapted to the Cauchy-Lagrangian method, such as the monitoring of the radius of convergence of the time-Taylor series. Certain other fluid equations can be handled similarly.

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“…We shall also address a new question: scaling exponents are notoriously known with poor accuracy (cf., e.g., [4]); how accurately can we determine such exponents by working with Reynolds number at which there are significant subdominant corrections to scaling? Using recent results of van der Hoeven [8,9], we shall show that this requires a subtle tradeoff between Reynolds numbers and precision (number of decimal digits) used in the calculations.We begin with simulation-based results for the Reynolds number dependence of gradmoments when standard double-precision calculations suffice. We follow here the same strategy as in Ref.[6]: we solve the Burgers equation (2) with the initial condition u 0 (x) = sin x, using a pseudo-spectral method combined with fixedtime-step fourth-order Runge-Kutta time marching and a slaved scheme, known by the acronym ETDRK4 [10], for handling the viscous dissipation.…”

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confidence: 91%

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confidence: 91%

“…The coefficients are given by rather complicated and numerically ill-conditionned integrals. The expansion (6) and the numerical values of the coefficients can actually be obtained by an alternative seminumerical procedure, called asymptotic extrapolation, developed by van der Hoeven [8] (see also [9] for an elementary presentation). Let us now say a few words about this technique, which will also be used below in connection with high-precision spectral calculations.…”

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confidence: 99%

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Nelkin scaling for the Burgers equation and the role of high-precision calculations

Chakraborty¹,

Frisch

Pauls

et al. 2012

Phys. Rev. E

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Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this procedure works already for Reynolds numbers of the order of 100 (or even lower when combined with a suitable extended self-similarity technique). At moderate Reynolds numbers, for the accurate determination of scaling exponents, it is crucial to use higher than double precision. Similar issues are likely to arise for three-dimensional Navier-Stokes simulations.

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A Borel Transform Method for Locating Singularities of Taylor and Fourier Series

Cited by 25 publications

References 41 publications

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

The Cauchy–Lagrangian method for numerical analysis of Euler flow

Nelkin scaling for the Burgers equation and the role of high-precision calculations

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