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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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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    High Order Numerical Discretization for Hamilton–Jacobi Equations on Triangular Meshes (2000)
    Springer
    Journal of scientific computing 15 (2000), S. 197-229 
    Details
    ISSN: 1573-7691
    Keywords: Hamilton–Jacobi equation ; viscosity solution ; numerical Hamiltonian ; $$\mathbb{P}^k $$ triangulation ; ε-monotonicity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract In this paper we construct several numerical approximations for first order Hamilton–Jacobi equations on triangular meshes. We show that, thanks to a filtering procedure, the high order versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are based on the first order Lax–Friedrichs scheme [2] which is improved here adjusting the dissipation term. The resulting first order scheme is ε-monotonic (we explain the expression in the paper) and converges to the viscosity solution as $$\mathcal{O}(\sqrt {\Delta t} )$$ for the L ∞-norm. The first high order method is directly inspired by the ENO philosophy in the sense where we use the monotonic Lax–Friedrichs Hamiltonian to reconstruct our numerical solutions. The second high order method combines a spatial high order discretization with the classical high order Runge–Kutta algorithm for the time discretization. Numerical experiments are performed for general Hamiltonians and L 1, L 2 and L ∞-errors with convergence rates calculated in one and two space dimensions show the k-th order rate when piecewise polynomial of degree k functions are used, measured in L 1-norm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007633810484
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  • 3
    Book
    Book
    Handbook of numerical methods for hyperbolic problems /; 17 (2016)
    Amsterdam [u.a.] :Elsevier, North-Holland,
    Details
    Title: Handbook of numerical methods for hyperbolic problems /; 17
    Contributer: Abgrall, Rémi , Shu, Chi-Wang
    Publisher: Amsterdam [u.a.] :Elsevier, North-Holland,
    Year of publication: 2016
    Pages: XXIII, 641 S. : , Illustrationen, Diagramme
    Series Statement: Handbook of numerical analysis 17
    ISBN: 978-0-444-63789-5
    ISSN: 1570-8659
    Type of Medium: Book
    Language: German , English
    URL: https://www.zbmath.org/?q=an:06657130
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  • 4
    Book
    Book
    Handbook of numerical methods for hyperbolic problems /; 18 (2017)
    Amsterdam [u.a.] :Elsevier, North-Holland,
    Details
    Title: Handbook of numerical methods for hyperbolic problems /; 18
    Contributer: Abgrall, Rémi , Shu, Chi-Wang
    Publisher: Amsterdam [u.a.] :Elsevier, North-Holland,
    Year of publication: 2017
    Pages: XIX, 589 S. : , Illustrationen, Diagramme
    Series Statement: Handbook of numerical analysis 18
    ISBN: 978-0-444-63910-3
    ISSN: 1570-8659
    Type of Medium: Book
    Language: German , English
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