Spectral Properties of Random and Almost Periodic Differential and Finite-Difference Operators

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

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Simulation of a seismic survey with combined surface and in-mine data acquisition for imaging steeply dipping ore veins

10.5194/egusphere-egu21-15350 ◽

2021 ◽

Author(s):

Olaf Hellwig ◽

Stefan Buske

Keyword(s):

Finite Difference ◽

Message Passing Interface ◽

Seismic Survey ◽

Difference Operators ◽

Mining District ◽

Prestack Depth Migration ◽

Data Set ◽

Depth Migration ◽

Difference Grid ◽

Triangulated Surfaces

<p>The polymetallic, hydrothermal deposit of the Freiberg mining district in the southeastern part of Germany is characterised by ore veins that are framed by Proterozoic orthogneiss. The ore veins consist mainly of quarz, sulfides, carbonates, barite and flourite, which are associated with silver, lead and tin. Today the Freiberg University of Mining and Technology is operating the shafts Reiche Zeche and Alte Elisabeth for research and teaching purposes with altogether 14 km of accessible underground galleries. The mine together with the most prominent geological structures of the central mining district are included in a 3D digital model, which is used in this study to study seismic acquisition geometries that can help to image the shallow as well as the deeper parts of the ore-bearing veins. These veins with dip angles between 40&#176; and 85&#176; are represented by triangulated surfaces in the digital geological model. In order to import these surfaces into our seismic finite-difference simulation code, they have to be converted into bodies with a certain thickness and specific elastic properties in a first step. In a second step, these bodies with their properties have to be discretized on a hexahedral finite-difference grid with dimensions of 1000 m by 1000 m in the horizontal direction and 500 m in the vertical direction. Sources and receiver lines are placed on the surface along roads near the mine. A Ricker wavelet with a central frequency of 50 Hz is used as the source signature at all excitation points. Beside the surface receivers, additional receivers are situated in accessible galleries of the mine at three different depth levels of 100 m, 150 m and 220 m below the surface. Since previous mining activities followed primarily the ore veins, there are only few pilot-headings that cut through longer gneiss sections. Only these positions surrounded by gneiss are suitable for imaging the ore veins. Based on this geometry, a synthetic seismic data set is generated with our explicit finite-difference time-stepping scheme, which solves the acoustic wave equation with second order accurate finite-difference operators in space and time. The scheme is parallelised using a decomposition of the spatial finite-difference grid into subdomains and Message Passing Interface for the exchange of the wavefields between neighbouring subdomains. The resulting synthetic seismic shot gathers are used as input for Kirchhoff prestack depth migration as well as Fresnel volume migration in order to image the ore veins. Only a top mute to remove the direct waves and a time-dependent gain to correct the amplitude decay due to the geometrical spreading are applied to the data before the migration. The combination of surface and in-mine acquisition helps to improve the image of the deeper parts of the dipping ore veins. Considering the limitations for placing receivers in the mine, Fresnel volume migration as a focusing version of Kirchhoff prestack depth migration helps to avoid migration artefacts caused by this sparse and limited acquisition geometry.</p>

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Grid Functions and Finite Difference Operators in 2D

Numerical Sound Synthesis ◽

10.1002/9780470749012.ch10 ◽

2009 ◽

pp. 287-304

Keyword(s):

Finite Difference ◽

Difference Operators

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Optimized staggered-grid finite-difference operators using window functions

Applied Geophysics ◽

10.1007/s11770-018-0668-7 ◽

2018 ◽

Vol 15 (2) ◽

pp. 253-260 ◽

Cited By ~ 2

Author(s):

Ying-Jun Ren ◽

Jian-Ping Huang ◽

Peng Yong ◽

Meng-Li Liu ◽

Chao Cui ◽

...

Keyword(s):

Finite Difference ◽

Staggered Grid ◽

Difference Operators ◽

Window Functions

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A review of recent work on almost periodic differential and difference operators

Acta Applicandae Mathematicae ◽

10.1007/bf00046601 ◽

1983 ◽

Vol 1 (3) ◽

pp. 241-261 ◽

Cited By ~ 8

Author(s):

Russell A. Johnson

Keyword(s):

Recent Work ◽

Difference Operators ◽

Almost Periodic

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Optimized Icosahedral Grids: Performance of Finite-Difference Operators and Multigrid Solver

Monthly Weather Review ◽

10.1175/mwr-d-12-00236.1 ◽

2013 ◽

Vol 141 (12) ◽

pp. 4450-4469 ◽

Cited By ~ 24

Author(s):

Ross P. Heikes ◽

David A. Randall ◽

Celal S. Konor

Keyword(s):

Finite Difference ◽

Convergence Analysis ◽

Optimization Methods ◽

Difference Operators ◽

Grid Spacing ◽

Multigrid Solver ◽

Poisson Equations ◽

The Earth ◽

Grid Optimization ◽

Accuracy Performance

Abstract This paper discusses the generation of icosahedral hexagonal–pentagonal grids, optimization of the grids, how optimization affects the accuracy of finite-difference Laplacian, Jacobian, and divergence operators, and a parallel multigrid solver that can be used to solve Poisson equations on the grids. Three different grid optimization methods are compared through an error convergence analysis. The optimization process increases the accuracy of the operators. Optimized grids up to 1-km grid spacing over the earth have been created. The accuracy, performance, and scalability of the multigrid solver are demonstrated.

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Solution of Structural Mechanics Problems by the Differential Quadrature Finite Difference Method

10.1115/imece2000-2461 ◽

2000 ◽

Author(s):

Chang-New Chen

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Difference Method ◽

Partial Differential Operator ◽

Structural Mechanics ◽

Plane Elasticity ◽

Differential Quadrature ◽

Difference Operators ◽

Elasticity Problems ◽

Nonuniform Plate

Abstract The differential quadrature finite difference method (DQFDM) has been proposed by the author. The finite difference operators are derived by the differential quadrature (DQ). They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy. By using different orders or the same order but different grid DQ discretizations for the same derivative or partial derivative, various finite difference operators for the same differential or partial differential operator can be obtained. Finite difference operators for unequally spaced and irregular grids can also be generated through the use of generic differential quadrature (GDQ). The derivation of higher order finite difference operators is also easy. By adopting the same order of approximation to all mathematical terms existing in the problem to be solved, excellent convergence can be obtained due to the consistent approximation. The DQFDM is effective for solving structural mechanics problems. The numerical simulations for solving anisotropic nonuniform plate problems and two-dimensional plane elasticity problems are carried out. Numerical results are presented. They demonstrate the DQFDM.

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