Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

Ranocha, Hendrik; Ostaszewski, Katharina; Heinisch, Philip

doi:10.1007/s42967-019-00057-2

Cited by 10 publications

(11 citation statements)

References 57 publications

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“…The legitimate question of orthogonality is discussed here by adopting the notations of figure (1) where n is the unit vector orthogonal to the surface S of the physical domain. Consider the two vectors f = ∇φ and g = ∇ × ψ and writing their inner product < f , g > given by orthogonal decomposition theorems [27], [28]:…”

Section: Boundary Conditions and Orthogonalitymentioning

confidence: 99%

On primitive formulation in fluid mechanics and fluid–structure interaction with constant piecewise properties in velocity–potentials of acceleration

Caltagirone

Vincent

2020

Acta Mech

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Discrete mechanics makes it possible to formulate any problem of fluid mechanics or fluid-structure interaction in velocity and potentials of acceleration; the equation system consists of a single vector equation and potentials updates. The scalar potential of the acceleration represents the pressure stress and the vector potential is related to the rotational-shear stress.The formulation of the equation of motion can be expressed in the form of a splitting which leads to an exact projection method; the application of the divergence operator to the discrete motion equation exhibits, without any approximation, a Poisson equation with constant coefficients on the scalar potential whatever the variations of the physical properties of the media. The a posteriori calculation of the pressure is made explicitly by introducing at this stage the local density.Two first examples show the interest of the formulation presented on classical solutions of Navier-Stokes equations; similarly as other results obtained with this formulation, the convergence is of order two in space and time for all the quantities, velocity and potentials. This formulation is then applied to a two-phase flow driven by surface tension and partial wettability. The last case corresponds to a fluid-structure interaction problem for which an analytical solution exists. J-P Caltagirone, S. Vincent, On primitive formulation in fluid mechanics and fluid-structure interaction with constant piecewise properties in velocity-potentials of acceleration, Acta Mechanica, 2020, https://doi.org/10.1007/s00707-020-02630-w ___________________________________________________________________________ interaction. However, the most delicate problem is posed for the Navier-Stokes equations which includes nonlinearities, which is not the case with the Navier-Lamé equation. The numerical treatment of Navier-Stokes equations in complex situations remains valid as is the theoretical order of the stability and convergence of the schemes used.The formulation proposed here is based on a physical model different from that of Navier-Stokes. It is briefly recalled later but the details of the derivation of the discrete equations are developed in [1]. The formulation based on the discrete mechanics allows access, at the same time, to the velocity V, compression stresses φ and shear ψ; this formulation (V, φ, ψ) can be written as a projection type algorithm by a splitting or be used as it is.Discrete geometric topologies are composed of a primal topology and a dual topology associated with specific variables where the scalars are located on the points of the primal geometry and the vectors on the edges of the latter. This vision is the generalization of the staggered Marker And Cell method on structured mesh. The polygonal or polyhedral meshes with any number of facets make it possible to pass over the usual 2D / 3D distinction for other methodologies.The operators of the primal and dual geometric topologies defined here have obvious analogies with those of DEC (Discrete Exterior Calc...

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Section: Boundary Conditions and Orthogonalitymentioning

confidence: 99%

On primitive formulation in fluid mechanics and fluid–structure interaction with constant piecewise properties in velocity–potentials of acceleration

Caltagirone

Vincent

2020

Acta Mech

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“…Nullspace consistency will be used as an additional required mimetic property. This novel property was introduced in [48] and has been a key factor in [18,38]. Here, 1 1 1 denotes the discrete grid function with value unity at every node.…”

Section: Remark 23mentioning

confidence: 99%

“…Therefore, The following technique has been used in [38] to analyze properties of SBP operators in space. Here, it will be used to create new SBP schemes in time.…”

Section: The New Schemesmentioning

confidence: 99%

A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

Ranocha

Nordström

2021

J Sci Comput

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Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.

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“…Hence, an automated approach using machine learning techniques was used instead (Ostaszewski et al, 2020). Due to their distinct shape and the comparatively large number of wave event occurrences it was possible to train a neural network to detect possible candidates for wave events with a high precision around 80%.…”

Section: Instrumentation and Event Selectionmentioning

confidence: 99%

Steepening of magnetosonic waves in the inner coma of comet 67P/Churyumov-Gerasimenko

Ostaszewski¹,

Glaßmeier²,

Goetz³

et al. 2020

Preprint

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Abstract. We present a statistical survey of large amplitude, asymmetric plasma, and magnetic field enhancements at comet 67P/Churyumov-Gerasimenko from December 2014 to June 2016. The aim is to provide a general overview of these structures' properties over the mission duration. At comets, nonlinear wave evolution plays an integral part in the development of turbulence and in particular facilitates the transfer of energy and momentum. As the first mission of its kind, the ESA Rosetta mission was able to study the plasma properties of the inner coma for a prolonged time and during different stages of activity. This enables us to study the temporal evolution of steepened waves and their characteristics. In total, we identified ~70000 events in the magnetic field data by means of machine learning. We observe that the occurrence of wave events is linked to the activity of the comet, where events are primarily observed at high outgassing rates. No clear indications of a relationship between the occurrence rate and solar wind conditions were found. The waves are found to propagate predominantly perpendicular to the background magnetic field, which indicates their compressive nature. Characteristics like amplitude, skewness, and width of the waves were extracted by fitting a skew normal distribution to the magnetic field magnitude of individual events. With increasing massloading the average amplitude of steepened waves decreases while the skewness increases. Using a modified 1D MHD model it was possible to show that such solitary structures can be described by the combination of nonlinear, dispersive, and dissipative effects. By combining the model with observations of amplitude, width, and skewness we obtain an estimate of the effective plasma viscosity in the comet-solar wind interaction region. At 67P/Churyumov-Gerasimenko steepened waves are of particular importance as they dominate the innermost interaction region for intermediate to high activity.

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Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

Cited by 10 publications

References 57 publications

On primitive formulation in fluid mechanics and fluid–structure interaction with constant piecewise properties in velocity–potentials of acceleration

On primitive formulation in fluid mechanics and fluid–structure interaction with constant piecewise properties in velocity–potentials of acceleration

A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

Steepening of magnetosonic waves in the inner coma of comet 67P/Churyumov-Gerasimenko

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