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  • 1
    Electronic Resource
    Electronic Resource
    On the use of Mühlbach expansions in the recovery step of ENO methods (1997)
    Springer
    Numerische Mathematik 76 (1997), S. 1-25 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65M20, 65M60, 65D05, 76M25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050252
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  • 2
    Electronic Resource
    Electronic Resource
    A dual graph-norm refinement indicator for finite volume approximations of the Euler equations (1998)
    Springer
    Numerische Mathematik 78 (1998), S. 619-658 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65M20, 65M50, 65M60; 76M99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We generalise and apply a refinement indicator of the type originally designed by Mackenzie, Süli and Warnecke in [15] and [16] for linear Friedrichs systems to the Euler equations of inviscid, compressible fluid flow. The Euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in [15] can be applied. We discuss the details of the implementation of the refinement indicator into the DLR- $\tau$ -Code which is based on a finite volume method of box type on an unstructured grid and present numerical results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050328
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  • 3
    Electronic Resource
    Electronic Resource
    Generalized multiresolution analysis on unstructured grids (2000)
    Springer
    Numerische Mathematik 86 (2000), S. 685-715 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65M55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Efficiency of high-order essentially non-oscillatory (ENO) approximations of conservation laws can be drastically improved if ideas of multiresolution analysis are taken into account. These methods of data compression not only reduce the necessary amount of discrete data but can also serve as tools in detecting local low-dimensional features in the numerical solution. We describe the mathematical background of the generalized multiresolution analysis as developed by Abgrall and Harten in [14], [15] and [3]. We were able to ultimately reduce the functional analytic background to matrix-vector operations of linear algebra. We consider the example of interpolation on the line as well as the important case of multiresolution analysis of cell average data which is used in finite volume approximations. In contrast to Abgrall and Harten, we develop a robust agglomeration procedure and recovery algorithms based on least-squeare polynomials. The efficiency of our algorithms is documented by means of several examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00005415
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  • 4
    Electronic Resource
    Electronic Resource
    On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions (1996)
    Springer
    Numerische Mathematik 74 (1996), S. 177-201 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65M60 41A15 41A29 41A63; 65N30 76M25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050213
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  • 5
    Electronic Resource
    Electronic Resource
    Regularising discretizations of the Birkhoff-Rott equation (1995)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 18 (1995), S. 615-625 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A thin shear layer moving from the trailing edge of a two-dimensional aerofoil section downstream can be interpreted as a curve of discontinuity for the tangential velocity and may be approximated by a vortex sheet in inviscid, incompressible fluid flow. It is well known that vortex sheets are subject to instabilities of Kelvin-Helmholtz type which lead to roll-up phenomena in the wake. The motion of such sheets is governed by the Birkhoff-Rott equation. In the case of Kelvin-Helmholtz instability it seems clear that a curvature singularity occurs at a certain critical time and that consistent discretizations of the Birkhoff-Rott equation may fail to yield reliable results even before the time of occurrence of a singularity. We discuss the modification of the Biot-Savart kernel in the sense of Krasny who regularized the kernel by means of a global parameter. Using discrete Fourier transform we show the damping influence of this regularization technique. We modify the kernel carefully by introducing a regularization found in ordinary vortex methods and show that reliable results may be obtained up to and slightly after the singularity formation without increasing the accuracy of the computation.
    Additional Material: 10 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670180804
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  • 6
    Electronic Resource
    Electronic Resource
    On the comparison of four different implementations of a third-order ENO scheme of box type for the computation of compressible flow (1996)
    Springer
    Numerical algorithms 13 (1996), S. 77-105 
    Details
    ISSN: 1572-9265
    Keywords: ENO recovery ; finite volume methods ; implicit time stepping ; compressible flow ; Primary: 65N40, 65D15, 76M20, 76M25 ; Secondary: 76N10, 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Following ideas of Abgrall, four different implementations of a third-order ENO scheme on general triangulations are described and examined. Two implementations utilize implicit time stepping where the resulting linear systems are solved by means of a preconditioned GMRES method. Two other schemes are constructed using an explicit Adams method in time. Quadratic polynomial recovery is used to result in a formally third-order accurate space discretisation. While one class of implementations makes use of cell averages defined on boxes and thus is close in spirit to the finite volume idea, the second class of methods considered is completely node-based. In this second case the interpretation as a true finite volume recovery is completely lost but the recovery process is much simpler and cheaper than the original one. Although one would expect a consistency error in the finite difference type implementations no such problem ever occurred in the numerical experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02143128
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  • 7
    Book
    Book
    Multivariate Rekonstruktionsverfahren zur numerischen Berechnung hyperbolischer Erhaltungsgleichungen. Zugl. Darmstadt, Technische Hochschule, Habil.-Schrift, 1994; 95-02 (1995)
    Köln :DLR,
    Details
    Title: Multivariate Rekonstruktionsverfahren zur numerischen Berechnung hyperbolischer Erhaltungsgleichungen. Zugl. Darmstadt, Technische Hochschule, Habil.-Schrift, 1994; 95-02
    Author: Sonar, Thomas
    Publisher: Köln :DLR,
    Year of publication: 1995
    Pages: 197 S.
    Series Statement: DLR Forschungsbericht 95-02
    Type of Medium: Book
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  • 8
    Book
    Book
    Angewandte Mathematik, Modellbildung und Informatik : eine Einführung für Lehramtsstudenten, Lehrer und Schüler (2001)
    Braunschweig u.a. :Vieweg,
    Details
    Title: Angewandte Mathematik, Modellbildung und Informatik : eine Einführung für Lehramtsstudenten, Lehrer und Schüler
    Author: Sonar, Thomas
    Publisher: Braunschweig u.a. :Vieweg,
    Year of publication: 2001
    Pages: 237 S.
    Type of Medium: Book
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