Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I

Лебедев, В. И.

doi:10.1016/0041-5553(64)90240-x

Cited by 87 publications

(53 citation statements)

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“…Similar consistent approximations of the vector analysis operators are most easy to realize for the simplest rectangular grids [33,34]. We note two classes of irregular grids: structured and unstructured grids.…”

Section: Introductionmentioning

confidence: 99%

Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

Vabishchevich¹

2005

Comput. Methods Appl. Math.

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-Mathematical physics problems are often formulated by means of the vector analysis differential operators: divergence, gradient and rotor. For approximate solutions of such problems it is natural to use the corresponding operator statements for the grid problems, i.e., to use the so-called VAGO (Vector Analys Grid Operators) method. In this paper, we discuss the possibilities of such an approach in using general irregular grids. The vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. The truncation error and the consistency property of the difference operators constructed on two types of grids are investigated. Construction and analysis of the difference schemes of the VAGO method for applied problems are illustrated by the examples of stationary and non-stationary convection-diffusion problems. The other examples concerned the solution of the nonstationary vector problems described by the second-order equations or the systems of first-order equations.2000 Mathematics Subject Classification: 65N06; 65M06.

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Section: Introductionmentioning

confidence: 99%

Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

Vabishchevich¹

2005

Comput. Methods Appl. Math.

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“…To numerically solve system (29), a method based on a finite difference scheme proposed in [24] and a decomposition method from [25] are used. We use a square 412 × 412 × 412 grid with a constant spacing.…”

Section: Numerical Simulationmentioning

confidence: 99%

The Subgrid Modeling for Maxwell's Equations With Multiscale Isotropic Random Conductivity and Permittivity

Курочкина¹,

Соболева²

2013

PIER B

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Abstract-The effective coefficients for Maxwell's equations in the frequency domain are calculated for a multiscale isotropic medium by using a subgrid modeling approach. The correlated fields of conductivity and permeability are approximated by Kolmogorov's multiplicative continuous cascades with a lognormal probability distribution. The wavelength is assumed to be large as compared with the scale of heterogeneities of the medium. The permittivity ε(x) and the electric conductivity σ(x) satisfy the condition σ(x)/(ωε(x)) < 1, where ω is the cyclic frequency. The theoretical results obtained in the paper are compared with the results from direct 3D numerical simulation.

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“…Virieux, 1986;Rojas et al, 2008) and fully staggered grids (FSG, e.g. Lebedev, 1964;de la Puente et al, 2014) resulting from updating their original Leap-Frog 2 explicit time-integration schemes to accomodate the mimetic approach. We use a two-dimensional representation for simplicity.…”

Section: Theorymentioning

confidence: 99%

Mimetic Operators for Seismic Exploration

Puente¹,

Rojas²,

Ferrer³

et al. 2016

78th EAGE Conference and Exhibition 2016

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SUMMARYMimetic operators are a kind of discrete operators which, on staggered grids, can accomplish "mimetic" properties analogous to their continuous counterparts. Furthermore, they can attain identical convergence order everywhere in the domain, including at or near the domain's boundaries in non-periodic problems. This property makes mimetic finite difference (MFD) operators attractive whenever high-contrast interfaces or physical boundary conditions are present in our models. In the seismic case, the strongest boundary condition is the interface between Earth and air, the so-called free surface. This surface is typically represented as a traction-free boundary condition and, although it is of critical importance for achieving accurate results, due to its vicinity to the sources and receivers, it is still a problem for current FD implementations. Despite efforts in the past, few FD schemes have attained convergence of order beyond two in results involving the free surface. Here we summarize the properties of mimetic operators applied to the elastic wave equation, as well as the changes required in order to incorporate into current explicit staggered grid codes the mimetic approach of the free-surface condition

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Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I

Cited by 87 publications

References 1 publication

Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

The Subgrid Modeling for Maxwell's Equations With Multiscale Isotropic Random Conductivity and Permittivity

Mimetic Operators for Seismic Exploration

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