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Elliptic coordinate system

About: Elliptic coordinate system is a(n) research topic. Over the lifetime, 670 publication(s) have been published within this topic receiving 11135 citation(s). The topic is also known as: elliptical coordinate system & elliptic coordinates.

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Abstract: A method for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected region containing any number of arbitrarily shaped bodies is presented. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regardless of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system. Numerical solutions for the lifting and nonlifting potential flow about Joukowski and Karman-Trefftz airfoils using this coordinate system generation show excellent comparison with the analytic solutions. The application to fields with multiple bodies is illustrated by a potential flow solution for multiple airfoils.

966 citations

Journal ArticleDOI
Abstract: A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.

529 citations

Journal ArticleDOI
Abstract: Description of a new method of writing the conservation equations of gasdynamics in curvilinear coordinates which eliminates undifferentiated terms. It is thus possible to readily apply difference schemes derived for Cartesian coordinates which conserve mass, momentum, and energy in the total flow field. The method is derived for orthogonal coordinates, and then extended to cover the most general class of coordinate transformations, using general tensor analysis. Several special features of the equations are discussed.

408 citations

Journal ArticleDOI
Abstract: The discrete variable representation (DVR) is used to calculate vibrational energy levels of H2O and SO2. The Hamiltonian is written in terms of bond length–bond angle coordinates and their conjugate momenta. It is shown that although these coordinates are not orthogonal and the appropriate kinetic energy operator is complicated, the discrete variable representation is quite simple and facilitates the calculation of vibrational energy levels. The DVR enables one to use an internal coordinate Hamiltonian without expanding the coordinate dependence of the kinetic energy or evaluating matrix elements numerically. The accuracy of previous internal coordinate calculations is assessed.

324 citations

Journal ArticleDOI
Abstract: The treatment of the geometrical singularity in cylindrical and spherical coordinates has for many years been a difficulty in the development of accurate finite difference (FD) and pseudo-spectral (PS) schemes. A variety of numerical procedures for dealing with the singularity have been suggested. For comparative purposes, some of these are discussed in the next sections, but the reader is referred to several books and review papers [3, 7, 10] for more detailed references. Generally, methods discussed in the literature use pole equations, which are akin to boundary conditions to be applied at the singular point. The treatment of the pole as a computational boundary can lead to numerical difficulties. These include the necessity of special boundary closures for FD schemes (e.g., [11]), undesirable clustering of grid points in PS schemes (e.g., [12]), and, in FD schemes, the generation of spurious waves which oscillate from grid point to grid point (so-called two-delta or sawtooth waves, see [4, 26]). In the present paper we investigate a method for treating the coordinate singularity whereby singular coordinates are redefined so that data are differentiated smoothly through the pole, and we avoid placing a grid point directly at the pole. This eliminates the need for any pole equation. Despite the simplicity of the present technique, it appears to be an effective and systematic way to treat many scalar and vector equations in cylindrical and spherical coordinates. A similar technique was used by Merilees [17] for the south and north pole singularities of spherical coordinates but appears not to have been applied more generally. Here we show that the technique leads to excellent results for a number of model problems. The main application we consider is the compressible unsteady Euler and Navier–Stokes equations in cylindrical coordinates (Section 3). For comparison with other methods, we also treat the solution of Bessel’s equation in Section 4. Several

292 citations

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