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  • 1
    Book
    Book
    Discontinuous Galerkin methods : theory, computation and applications; 11 (2000)
    Berlin u.a. :Springer,
    Details
    Title: Discontinuous Galerkin methods : theory, computation and applications; 11
    Contributer: Cockburn, Bernardo , Karniadakis, George E. , Shu, Chi-Wang
    Publisher: Berlin u.a. :Springer,
    Year of publication: 2000
    Pages: 470 S.
    Series Statement: Lecture notes in computational science and engineering 11
    Type of Medium: Book
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  • 2
    Book
    Book
    Advanced numerical approximation of nonlinear hyperbolic equations : lectures given at the 2nd session of the Centro Internationale Matematico Estivo (C.I.M.E), Cetraro, Italy, 1997; 1697 (1998)
    Berlin u.a. :Springer,
    Details
    Title: Advanced numerical approximation of nonlinear hyperbolic equations : lectures given at the 2nd session of the Centro Internationale Matematico Estivo (C.I.M.E), Cetraro, Italy, 1997; 1697
    Author: Cockburn, Bernardo
    Contributer: Johnson, Claes , Shu, Chi-Wang , Tadmor, Eitan , Quarteroni, Alfio
    Publisher: Berlin u.a. :Springer,
    Year of publication: 1998
    Pages: 450 S.
    Series Statement: Lecture notes in mathematics 1697
    Type of Medium: Book
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  • 3
    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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  • 4
    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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  • 5
    Electronic Resource
    Electronic Resource
    On positivity preserving finite volume schemes for Euler equations (1996)
    Springer
    Numerische Mathematik 73 (1996), S. 119-130 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65M60, 76M20, 76K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional Euler equations of gas dynamics. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve a high order positivity preserving reconstruction is described.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050187
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  • 6
    Electronic Resource
    Electronic Resource
    On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function (1995)
    Springer
    Numerische Mathematik 71 (1995), S. 511-526 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 42A15, 41A05, 41A25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. This paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponentially convergent approximation to the function $f(x)$ in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods. A numerical example is also provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050155
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  • 7
    Electronic Resource
    Electronic Resource
    Effective equations and the inverse cascade theory for Kolmogorov flows (1993)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 5 (1993), S. 998-1010 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: A detailed study is presented of the inverse cascade process in the Kolmogorov flows in the limit as the forcing wave number goes to infinity. Extensive numerical results reveal that energy is transferred from high wave numbers to low ones through two distinctive stages. At the early stage, the low wave-number spectrum evolves to a universal k−4 decay law. At the same time, enough scales are generated between the forcing scale (small) and the size of the system (large scale) so that there is no longer separation of scales in the system. In the next stage the flow undergoes a transition to a continuum of scales. This transition proceeds through the formation of a hierarchy of layers of elongated vortices at increasingly larger scales with alternating orientation. To explain these phenomena, effective equations governing the evolution of the large-scale quantities are derived. Strong numerical evidence is presented that even with smooth initial data, the solution to the effective equation develops a k−4 type singularity at a finite time. The effective equation also exhibits a weak instability which suggest that gradients in the direction of the forcing will grow much quicker than gradients in other directions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.858644
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  • 8
    Electronic Resource
    Electronic Resource
    Small-scale structures in Boussinesq convection (1994)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 6 (1994), S. 49-58 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Two-dimensional Boussinesq convection is studied numerically using two different methods: a filtered pseudospectral method and a high-order accurate eno scheme. The issue whether finite time singularity occurs for initially smooth flows is investigated. In contrast to the findings of Pumir and Siggia who reported finite time collapse of the bubble cap, the present numerical results suggest that the strain rate corresponding to the intensification of the density gradient across the front saturates at the bubble cap. Consequently, the thickness of the bubble decreases exponentially. On the other hand, the bubble experiences much stronger straining and intensification of gradients at its side. As the bubble rises, a secondary front also forms from its tail. Together with the primary front, they constitute a pair of tightly bound plus and minus double vortex sheet structure which is highly unstable and vulnerable to viscous dissipation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.868044
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  • 9
    Electronic Resource
    Electronic Resource
    A note on the accuracy of spectral method applied to nonlinear conservation laws (1995)
    Springer
    Journal of scientific computing 10 (1995), S. 357-369 
    Details
    ISSN: 1573-7691
    Keywords: Spectral method ; accuracy ; Gibbs phenomenon ; nonlinear conservation laws
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear PDE with discontinuous solutions, Fourier spectral method will produce poor point-wise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this note we assess the accuracy of Fourier spectral method applied to nonlinear conservation laws through a numerical case study. We have found out that the moments against analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a Gegenbauer polynomial based post-processing.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02091780
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  • 10
    Electronic Resource
    Electronic Resource
    Numerical experiments on the accuracy of ENO and modified ENO schemes (1990)
    Springer
    Journal of scientific computing 5 (1990), S. 127-149 
    Details
    ISSN: 1573-7691
    Keywords: ENO schemes ; accuracy ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract In this paper we make further numerical experiments assessing an accuracy degeneracy phenomena reported by A. Rogerson and E. Meiburg (this issue, 1990). We also propose a modified ENO scheme, which recovers the correct order of accuracy for all the test problems with smooth initial conditions and gives results comparable to the original ENO schemes for discontinuous problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01065581
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