Design of an Essentially Nonoscillatory Reconstruction Procedure on Finite-Element-Type Meshes

Abgrall, Rémi

doi:10.1007/978-3-642-60543-7_21

Cited by 4 publications

(5 citation statements)

References 11 publications

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“…This is, in all shortness, the essence of essentially non-oscillatory (ENO) recovery algorithms which were designed by Harten and his co-workers for the one-dimensional case in a series of papers, see [5], [6], [7], [8]. A multi-dimensional analogon of the ENO theory was provided in [1], [2] using finite element interpolation theory. Here, the π i 's are m-vectors of polynomials of fixed degree on the boxes σ i .…”

Section: Essentially Non-oscillatory Polynomial Recoverymentioning

confidence: 99%

“…According to the theory developed in [1], [2] the final quadratic recovery function π i is the one for which the sum of the absolut values of the leading coefficients is minimised, i.e.…”

Section: Essentially Non-oscillatory Polynomial Recoverymentioning

confidence: 99%

“…Thus, we consider linear finite element hat functions on the triangle spanned by x 1 , x 2 , x 3 interpolating Φ at these points. We would expect the interpolant to be the function Φ(x 1 )ϕ 1 …”

Section: An Interpolatory Examplementioning

confidence: 99%

“…As shown in [2], the barycentric coordinates introduced in Sect. 3.4 are good candidates since it can be shown that the conditioning number of the linear systems are independent of the mesh size, contrary to what happens in case of a Taylor expansion, see [1]. For that result to hold true one has to assume some technical conditions.…”

Section: Compressible Ramp Flowmentioning

confidence: 99%

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On the use of Mühlbach expansions in the recovery step of ENO methods

Abgrall

Sonar²

1997

Numerische Mathematik

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The recovery step is the most expensive algorithmic ingredient in modern essentially non-oscillatory (ENO) shock capturing methods on triangular meshes for the numerical simulation of compressible fluid flow. While recovery polynomials in Newton form are used in one-dimensional ENO schemes it is a priori not clear whether such useful as well as numerically stable form of polynomials exists in multiple dimensions. As was observed in [1] a very general answer to this question was provided by Mühlbach in two subsequent papers [15] and [16]. We generalise his interpolation theory further to the general recovery problem and outline the use of Mühlbach's expansion in ENO schemes. Numerical examples show the usefulness of this approach in the problem of recovery from cell average data. Subject Classification (1991): 65M20, 65M60, 65D05, 76M25 Mathematics

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Section: Essentially Non-oscillatory Polynomial Recoverymentioning

confidence: 99%

“…According to the theory developed in [1], [2] the final quadratic recovery function π i is the one for which the sum of the absolut values of the leading coefficients is minimised, i.e.…”

Section: Essentially Non-oscillatory Polynomial Recoverymentioning

confidence: 99%

Section: An Interpolatory Examplementioning

confidence: 99%

Section: Compressible Ramp Flowmentioning

confidence: 99%

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On the use of Mühlbach expansions in the recovery step of ENO methods

Abgrall

Sonar²

1997

Numerische Mathematik

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“…Unfortunately, such stencils can hamper convergence to steady state. Also, ENO schemes are not easily implemented on unstructured meshes 6,7].…”

Section: Introductionmentioning

confidence: 99%

A new class of ENO schemes based on unlimited data-dependent least-squares reconstruction

Ollivier‐Gooch

1996

34th Aerospace Sciences Meeting and Exhibit

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A crucial step in obtaining high-order accurate steady-state solutions to the Euler and Navier-Stokes equations is the high-order accurate reconstruction of the solution from cell-averaged values. Only after this reconstruction has been completed can the ux integral around a control volume be accurately assessed. In this work, a new reconstruction scheme is presented that is conservative, uniformly accurate with no overshoots, easy to implement on arbitrary meshes, has good convergence properties, and is computationally e cient. The new scheme, called DD-L 2 , uses a data-dependent weighted least-squares reconstruction with a xed 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 can be used. The new scheme satis es a relaxed version of the ENO criteria. Local accuracy of the reconstruction is optimal except for functions that are continuous but have discontinuous low-order derivatives. The total variation of the reconstruction is bounded by the total variation of the function to within O (x). The asymptotic behavior of the scheme in reconstructing smooth and piecewise smooth functions is demonstrated. DD-L 2 produces uniformly highorder accurate reconstructions, even in the presence of discontinuities. Two-dimensional ow solutions obtained using DD-L 2 reconstruction are compared with solutions using limited least-squares reconstruction. The solutions are virtually identical. The ab

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