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Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.


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Journal ArticleDOI
TL;DR: In this article, the authors give convergence criteria for general difference schemes for boundary value problems in Lipschitzian regions, and prove convergence for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.
Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.

93 citations

Journal ArticleDOI
TL;DR: In this article, two moving mesh partial differential equations (MMPDEs) with spatial smoothing are derived based upon the equidistribution principle, motivated by the robust moving mesh method of Dorfi and Drury.
Abstract: Two moving mesh partial differential equations (MMPDEs) with spatial smoothing are derived based upon the equidistribution principle. This smoothing technique is motivated by the robust moving mesh method of Dorfi and Drury [J. Comput. Phys., 69 (1987), pp. 175--195]. It is shown that under weak conditions the basic property of no node-crossing is preserved by the spatial smoothing, and a local quasi-uniformity property of the coordinate transformations determined by these MMPDEs is proven. It is also shown that, discretizing the MMPDEs using centered finite differences, these basic properties are preserved.

93 citations

Journal ArticleDOI
TL;DR: In this article, the properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated, and the velocity is expanded in terms of poloidal and toroidal vectors.
Abstract: The properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated. All nonlinear terms are retained in the equations, and both axisymmetric and nonaxisymmetric solutions are studied. The velocity is expanded in terms of poloidal and toroidal vectors. Spherical surface harmonics resolve the horizontal structure of the flow, but finite differences are used in the vertical. With a few modifications, the transform method developed by Orszag (1970) is used to calculate the nonlinear terms, while Green's function techniques are applied to the poloidal equation and diffusion terms.

93 citations

Journal ArticleDOI
TL;DR: In this article, the 3D finite difference method is used to simulate borehole wave propagations in an isotropic as well as an anisotropic formation, and the finite difference results agree excellently with the analytic solutions of a point force source in the transversely isropic medium.
Abstract: In this paper the three‐dimensional finite difference method is used to simulate borehole wave propagations in an isotropic as well as an anisotropic formation. The finite difference results agree excellently with the analytic solutions of a point force source in the transversely isotropic medium. The finite difference synthetics are also in very good agreement with the discrete wave‐number solutions for fluid‐filled borehole wave propagation. The finite difference synthetics are compared with ultrasonic lab measurements in a scaled borehole model. The borehole is drilled along the X axis in an orthorhombic phenolite solid. Both monopole and dipole logs agree well. The observations of the shear wave splitting in the dipole logs are confirmed by the finite difference simulations. The 3‐D finite difference method is applied to the fluid‐filled borehole wave propagation in the tilted isotropic formation and in the orthorhombic phenolite formation. In a borehole drilled along the Z axis in a phenolite formati...

93 citations

Book ChapterDOI
01 Jan 1973
TL;DR: The analysis of crack problems in plane elasticity has intrigued mathematicians for nearly sixty years and many mathematical approaches with wide ranges of sophistication have been applied to a variety of crack configurations and loading conditions as mentioned in this paper.
Abstract: The analysis of crack problems in plane elasticity has intrigued mathematicians for nearly sixty years. Inglis [1] found the solution for a single crack in an infinite sheet with the use of elliptic coordinates. Since then, many mathematical approaches with wide ranges of sophistication have been applied to a variety of crack configurations and loading conditions. It is easy to appreciate the mathematical interest in a problem area in which solution techniques span such diverse topics as analytic function theory, integral equations, transform methods, conformal mapping, boundary collocation, finite differences, finite elements, asymptotic methods, etc.

93 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023153
2022411
2021722
2020679
2019678
2018708