T. J. Barth and P. O. Frederickson, Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA Paper 90-0013, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the coe#cients have a complex nonlinear dependence on the solution. 3.4.2. Artificial Di#usion and Upwinding. Following the pioneering work of Godunov [60] a variety of dissipative and upwind schemes designed to have good shock capturing properties have been developed during the past two decades [182, 27, 107, 108, 163, 142, 65, 141, 185, 8, 79, 204, 73, 201, 16, 15, 17]. If the one dimensional scalar conservation law #x f(v) 0 (9) is represented by a three point scheme = c v j ) c (v j 1 v j ) the scheme is LED if 0,c 0. 10) A conservative semidiscrete approximation to the one dimensional conservation law can be derived by ....

....reconstruction. An example is the class of essentially non oscillatory (ENO) schemes, which can attain a very high order of accuracy at the cost of a substantial increase in computational complexity [37, 170, 168, 169] Methods based on reconstruction can also be implemented on unstructured meshes [16, 15]. Recently there has been an increasing interest in kinetic flux splitting schemes, which use solutions of the Boltzmann equation or the BGK equation to predict the interface fluxes [50, 41, 54, 153, 202] 3.5. Multidimensional Schemes. The simplest approach to the treatment of multi dimensional ....

T. Barth and P. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA paper 90-0013, AIAA 28th Aerospace Sciences Meeting, January 1990.

....grid topologies such as triangular tetrehedral [9] tree based adaptive [10] and arbitrary connectivity polygonal grids [11] pose significant problems for schemes using simple extensions of onedimensional methods. Our starting point are multi dimensional k exact methods devised by Barth [12, 13, 14, 15, 16]. We specifically embrace Barth s approach for deriving a reconstruction based upon least squares methodology [17] Barth applies monotonicity after the minimization process, following principles similar to those set forth by Dukowicz [18, 19] and Zalesak in his multi dimensional FCT algorithm ....

T. J. Barth and P. O. Frederickson. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. In 28th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 1990. AIAA--90--0013.

.... (though we have not observed this, a neighborhood that is nearly wellconditioned might switch back and forth between different sets of basis functions) Better would be to consistently use the same set of basis functions and optimize some auxiliary norm in the underdetermined case, as in Barth [1]. This requires an orthogonal decomposition of S T S, an added computational expense that degrades the overall interactivity of the modeler. 3.3 Curve reconstruction Smooth neighborhoods for PWL curve approximations are computed analogously to the fitting procedure for surfaces. A chordlength ....

Timothy Barth. Higher order solution of the euler equations on unstructured grids using quadratic reconstruction. In 28th Aerospace Sciences Meeting. AIAA-90-0013, 1990.

....grid with an unstructured cellbased data structure. Initial grid generation is enhanced by geometry based refinement [38] Solution adaptation is achieved by solution based gradient information [40, 64] The flow solver is based on the MUSCL concept [60] with a linear reconstruction technique [4], and either Roe s approximate Riemann solver [50] or Van Leer s flux splitting [30] The solution is advanced in time by a multi stage time stepping scheme [54, 62] with multigrid convergence acceleration [9] The data structure and grid generation are discussed in Chapter II. Each element of ....

....approximation of the solution on each cell for the projection stage leads to secondorder spatial accuracy, while a quadratic representation on each cell leads to thirdorder spatial accuracy. In this work, a second order spatial accuracy is achieved by a linear reconstruction procedure on each cell [4]. The effect on UL and UR by this higher order reconstruction is pictured in Figure 3.2. Unfortunately, limiting is required to enforce the monotonicity of the reconstruction. Wherever limiting is used, the spatial accuracy of the projection is somewhat reduced, possibly to a first 47 48 L R ....

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T. J. Barth and P. O. Frederickson, "Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction," AIAA Paper 90-0013, 1990.

....to offer more promise. However, most of the relatively new family of fluctuation splitting cell vertex schemes (cf. 14, 38] are nonlinear, with the linear LDA scheme rarely used being in practice. Darmofal [12] has derived the Fourier footprint for Barth s k = 0 (first order) upwind scheme [4] for the 2 D convection equation on a triangular mesh. Assuming that there are six equilateral triangles meeting in a node, it is possible to define Fourier variables fi 1 , fi 2 and fi 3 in the directions of the edges radiating from the 137 interior node. Only two of these variables are ....

T. J. Barth and P. O. Frederickson, "Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction," AIAA Paper 90-0013, 1990.

....feature of unstructured versus structured meshes in a reconstruction context. Both limited k exact reconstruction and essentially non oscillatory (ENO) reconstruction have many of these desirable properties. The design criteria for k exact reconstruction given by Barth and Frederickson [1] ensure conservation of the mean, good accuracy for smooth functions, and computational efficiency. These schemes are easy to implement on arbitrary meshes. A fixed stencil is used, allowing pre computation of certain purely geometric quantities used in the reconstruction, with a corresponding ....

....that has the best properties of both limited k exact reconstruction schemes and ENO schemes. The new scheme, called DD L 2 ENO, uses a data dependent weighted least squares reconstruction with a fixed stencil. To this extent, the scheme resembles existing k exact reconstruction algorithms [1, 10] and can build on that technology base. The feature which distinguishes DD L 2 ENO from 4 standard k exact schemes is the use of data dependent weights. Each point in a reconstruction stencil is given a data dependent weight, chosen to strongly emphasize smooth data in the reconstruction. ....

Barth, T. J. and Frederickson, P. O., "Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction." AIAA paper 90-0013, Jan. 1990.

....2.4.1 Minimum Energy Reconstruction The cell primitive variables in each cell are reconstructed, in the spirit of MUSCL interpolation [79] using a linear reconstruction procedure. The reconstruction used is based upon the Minimum Energy reconstruction presented by Barth and Frederickson in [4]. The process of reconstruction to arbitrary degrees of accuracy using this approach is presented in [4] 5] 6] and [20] A similar interpretation is presented below. Reconstruction can be viewed as being the discrete inverse of the cell averaging process. Define the cell average of an arbitrary ....

....in the spirit of MUSCL interpolation [79] using a linear reconstruction procedure. The reconstruction used is based upon the Minimum Energy reconstruction presented by Barth and Frederickson in [4] The process of reconstruction to arbitrary degrees of accuracy using this approach is presented in [4][5] 6] and [20] A similar interpretation is presented below. Reconstruction can be viewed as being the discrete inverse of the cell averaging process. Define the cell average of an arbitrary function, u, to be (2.1) The reconstruction solves the inverse of the cell averaging process: find the ....

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T. J. Barth and P. O. Frederickson. Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction. AIAA Paper 90-0013, 1990.

....method that can be enriched in a compatible manner by both finite volume or finite element discretized extra terms such as diffusion terms. This should give a better understanding of the derivation of several existing schemes for which numerical experiments are available (see [2] 3 8, 21] 10] [11]) for upwind schemes and [14, 22] for FCT schemes and in the other references given in the text. 1 Mathematical models We shall consider 2D situations for the sake of simplicity but most of the ideas presented in this note can be extended to 3D. We are interested by schemes that apply to a ....

T.J. BARTH, P.O. FREDERICKSON, "Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction", AIAA paper, 28th Aerospace Sciences Meeting, January 8--11, 1990/Reno, Nevada.

....of mesh points is much more dicult to choose; and, secondly, the Vandemonde determinants that arise in constructing the Newton fundamental polynomials are very badly conditioned. Durlofski et al. 13] and Abgrall [1] in extending ENO scheme ideas to unstructured meshes, and Barth and Frederickson [3], in making use of least squares criteria, have mitigated these diculties to some extent. But we will adopt here a much more direct approach to generalising the formulae of section 2.2 and 2.3. Discontinuous linear recovery, which is conservative in the mean and thus generalises (1.7) or (A.15) ....

T. J. Barth and P. O. Frederickson. Higher order solution of the Euler equations on unstructured grids using quadratic interpolation. AIAA 28th Aerospace Science Meeting, Paper 90-0013, 1990.

....for a locally uniform and a locally non uniform situation. We use the concept of reconstruction of piecewise C 1 functions from the cell wise constant data, q l i;j , associated with each cell. This idea was introduced in [21] and [24] for one dimensional convection and extended and applied in [1] and [2] for unstructured grids in two spatial dimensions. Contrary to this work, we do not reconstruct a single, unique (vector) function in each cell, but we take care that in a locally non uniform grid situation the computation of the left and right state is so that the resulting scheme is ....

T.J. Barth and P.O. Frederickson. Higher order solution of the Euler equations on unstructered grids, using quadratic reconstruction. In AIAA-90-0013, 1990.

....on arbitrary meshes, have good convergence properties, and be computationally efficient. Both limited k exact reconstruction and essentially non oscillatory (ENO) reconstruction have many of these properties. For example, design criteria for k exact reconstruction given by Barth and Frederickson [1] ensure conservation of the mean, good accuracy for smooth functions, and computational efficiency. A fixed stencil is used, allowing precomputation of certain purely geometric quantities used in the reconstruction, with a corresponding improvement in efficiency. Accuracy near discontinuities is ....

....least squares weighted reconstruction with a fixed stencil. The weights are chosen to strongly emphasize smooth data in the reconstruction. Because DD L 2 is designed in the framework of k exact reconstruction, existing techniques for implementing such reconstructions on arbitrary meshes [1, 8] can be used with only slight modification. The new scheme satisfies a relaxed version of the ENO criteria. Local accuracy of the reconstruction is optimal except in the case of functions that are continuous but have discontinuous low order derivatives. The total variation of the reconstruction is ....

[Article contains additional citation context not shown here]

Barth, T. J. and Frederickson, P. O., "Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction." AIAA paper 90-0013, Jan. 1990.

....generated by marching out procedures [93, 67] On a given grid, one has the option of locating the variables at the cell centers or at the vertices of the grid, giving rise to cell centered and cell vertex schemes. Alternatively, it is possible to deal strictly with averages defined over volumes [11, 27]. This approach has certain advantages for higher order schemes, but is not considered in the present work. In the case of finite volume schemes, the governing equations in integral form (Eqn. 1) are discretized. This allows discontinuities to be captured as part of the solution. Eqn. 1) ....

....deviate from the concept of cell averages. This would require that the reconstruction procedure only make use of cell averages and not point values. This is an attractive proposition for higher order schemes. Higher order accurate schemes based cell averages have been proposed and tested in [11, 27, 55]. In the following, we will review explicit schemes for unstructured grids and some acceleration techniques. The techniques of residual averaging and temporal adaptation that relax time step restrictions are briefly reviewed. Finally, the development of implicit schemes that allow for arbitrarily ....

T. J. Barth and P. O. Fredrickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper 90-0013, Jan. 1990.

....on arbitrary meshes, have good convergence properties, and be computationally efficient. Both limited k exact reconstruction and essentially non oscillatory (ENO) reconstruction have many of these properties. For example, design criteria for k exact reconstruction given by Barth and Frederickson [1] ensure conservation of the mean, good accuracy for smooth functions, and computational efficiency. A fixed stencil is used, allowing precomputation of certain purely geometric quantities used in the reconstruction, with a corresponding improvement in efficiency. Accuracy near discontinuities is ....

....least squares weighted reconstruction with a fixed stencil. The weights are chosen to strongly emphasize smooth data in the reconstruction. Because DD L 2 is designed in the framework of k exact reconstruction, existing techniques for implementing such reconstructions on arbitrary meshes [1, 8] can be used with only slight modification. The new scheme satisfies a relaxed version of the ENO criteria. Local accuracy of the reconstruction is optimal except in the case of functions that are continuous but have discontinuous low order derivatives. The total variation of the reconstruction is ....

[Article contains additional citation context not shown here]

Barth, T. J. and Frederickson, P. O., "Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction. " AIAA paper 90-0013, Jan. 1990.

No context found.

T. J. Barth and P. O. Frederickson, Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA Paper 90-0013, 1990.

No context found.

T. J. Barth and P. O. Fredrickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper 90-0013, Jan. 1990.

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T. J. Barth and P.O.Frederickson, Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA Paper 90-0013, 1990.

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Barth, T. J., and Frederickson, P. O., "Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction", AIAA-90-0013, Jan. 8-11, 1990.

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