The Cauchy–Lagrangian method for numerical analysis of Euler flow

Podvigina, Olga; Zheligovsky, Vladislav; Frisch, U.

doi:10.1016/j.jcp.2015.11.045

Cited by 28 publications

(35 citation statements)

References 37 publications

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“…(2011) for related discussions within the spherical collapse model, and Podvigina et al (2016) for highly related techniques for incompressible Euler flow. In the present case, at least for the displacement, it is a priori not ruled if the first singularity occurs for real times.…”

Section: Spherical Case: Convergence Until Collapsementioning

confidence: 99%

Quasi-spherical collapse of matter in ΛCDM

Rampf

2019

Monthly Notices of the Royal Astronomical Society

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We report the findings of new exact analytical solutions to the cosmological fluid equations, namely for the case where the initial conditions are perturbatively close to a spherical tophat profile. To do so we enable a fluid description in a Lagrangian-coordinates approach, and prove the convergence of the Taylor-series representation of the Lagrangian displacement field until the time of collapse ("shell-crossing"). This allows the determination of the time for quasi-spherical collapse, which is shown to happen generically earlier than in the spherical case. For pedagogical reasons, calculations are first given for a spatially flat universe that is only filled with a non-relativistic component of cold dark matter (CDM). Then, the methodology is updated to a ΛCDM Universe, with the inclusion of a cosmological constant Λ > 0.

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Section: Spherical Case: Convergence Until Collapsementioning

confidence: 99%

Quasi-spherical collapse of matter in ΛCDM

Rampf

2019

Monthly Notices of the Royal Astronomical Society

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“…It must be noted that mathematically they are perfectly suitable for numerical work: indeed, the presence of the inverse Laplacian in the second relation (19) suggests, that on increasing the number n of the coefficient s (n) k they become smoother and their energy spectrum decay is steeper. This is in sharp contrast, for instance, with the recurrence relations for the Lagrangian time-Taylor coefficients in the expansions of solutions to the Euler equation for incompressible fluid flow [23].…”

Section: Padé Approximationmentioning

confidence: 90%

Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation

Gama

Chertovskih

Zheligovsky

2019

Fluids

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We present examples of Padé approximation of the α-effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow, where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore application of Padé approximants for computation of tensors of magnetic α-effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Padé approximants of the tensors expanded in power series in the inverse molecular diffusivity 1/η around 1/η = 0. This yields the values of the dominant growth rate due to the action of the α-effect or eddy diffusivity to satisfactory accuracy for η, several dozen times smaller than the threshold, above which the power series is convergent. For one sample flow, we observe eddy diffusivity tending to negative infinity when η tends from above to the point of the onset of small-scale dynamo action in a symmetry-invariant subspace where a neutral small-scale magnetic mode resides. However, 49 first coefficients in the power series in 1/η prove insufficient for Padé approximants to reproduce this behaviour. We do computations in Fortran in the standard "double" (real*8) and extended "quadruple" (real*16) precision, as well as perform symbolic calculations in Mathematica. (Sílvio M.A. Gama) unity. For finite-precision computations, however, the cancellation is not any more guaranteed, and the initial growth of individual terms can result in ultimate loss of accuracy.The other one stems from finiteness of the radius of convergence of most power series encountered in computational practice. A complementary technique is then needed to continue analytically a function defined by the power series outside its circle of convergence. This can be achieved by constructing the so-called Padé approximants [14,4,5], i.e., an implementation of the continuation in the form of the ratio of two polynomials. Let us cite the words of appraisal in [24]: "Padé approximation has the uncanny knack of picking the function you had in mind from among all the possibilities. Except when it doesn't! That is the downside of Padé approximation: it is uncontrolled. There is, in general, no way to tell how accurate it is, or how far out in x it can usefully be extended. It is a powerful, but in the end still mysterious, technique."A not less mysterious notion is that of eddy diffusivity [28], also known as eddy (or turbulent) viscosity when fluid viscosity, the source of diffusion in hydrodynamics, is considered. Like the magnetic α-effect and anisotropic kinetic alpha-(aka AKA) effect, eddy diffusivities are often encountered in magnetohydrodynamics when generation of large-scale magnetic fields by flows of electrically conductive fluids is considered. At first sight, i...

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“…There are several ways in which the full nonlinear ideal MHD equations (compressible or incompressible) can be recast as Lie-advection problems, leading to Cauchy invariants equations. It is however not clear at the moment if such formulations lead to interesting results on time-analyticity and numerical integration by Cauchy-Lagrange-type methods Podvigina et al 2016). Similar questions arise for the extended MHD models discussed in Section 2.5.4.…”

Section: Conclusion and Open Problemsmentioning

confidence: 96%

Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces

Besse

Frisch

2017

J. Fluid Mech.

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Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of threedimensional (3D) ideal flow Podvigina et al. 2016). Looking at such invariants with the modern tools of differential geometry and of geodesic flow on the space SDiff of volume-preserving transformations (Arnold 1966), all manners of generalisations are here derived. The Cauchy invariants equation and the Cauchy formula, relating the vorticity and the Jacobian of the Lagrangian map, are shown to be two expressions of this Lie-advection invariance, which are duals of each other (specifically, Hodge dual). Actually, this is shown to be an instance of a general result which holds for flow both in flat (Euclidean) space and in a curved Riemannian space: any Lie-advection invariant p-form which is exact (i.e. is a differential of a (p − 1)-form) has an associated Cauchy invariants equation and a Cauchy formula. This constitutes a new fondamental result in linear transport theory, providing a Lagrangian formulation of Lie advection for some classes of differential forms. The result has a broad applicability: examples include the magnetohydrodynamics (MHD) equations and various extensions thereof, discussed by Lingam, Milosevich & Morrison (2016) and include also the equations of Tao (2016), Euler equations with modified Biot-Savart law, displaying finite-time blow up. Our main result is also used for new derivations -and several new results -concerning local helicity-type invariants for fluids and MHD flow in flat or curved spaces of arbitrary dimension.

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The Cauchy–Lagrangian method for numerical analysis of Euler flow

Cited by 28 publications

References 37 publications

Quasi-spherical collapse of matter in ΛCDM

Quasi-spherical collapse of matter in ΛCDM

Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation

Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces

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