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  • 1
    Electronic Resource
    Electronic Resource
    The group velocity of some numerical schemes (1985)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 5 (1985), S. 201-224 
    Details
    ISSN: 0271-2091
    Keywords: Numerical Group Velocity ; Fourier Analysis ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The performances of various numerical schemes used to model hyperbolic/parabolic equations have been studied by the calculation of their numerical group velocities. Numerical experiments conducted with one dimensional linear and quadratic Lagrangian finite elements with a Crank-Nicolson finite differencing in time confirm the results of the analysis. The group velocity analysis supplements the well-known amplitude and phase portraits introduced by Leendertse1 and helps explain the occurrence and behaviour of numerical oscillations in both finite difference and finite element schemes.
    Additional Material: 16 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650050302
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Internal wave reflections and transmissions arising from a non-uniform mesh. Part III: The occurrence of evanescent waves in the Crank-Nicolson linear finite element scheme (1989)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 9 (1989), S. 833-853 
    Details
    ISSN: 0271-2091
    Keywords: Fourier analysis ; Dispersion relation ; Reflected/transmitted evanescent waves ; Crank-Nicolson linear finite element scheme ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The numerical scheme upon which this paper is based is the 1D Crank-Nicolson linear finite element scheme. In Part I of this series it was shown that for a certain range of incident wavelengths impinging on the interface of an expansion in nodal spacing, an evanescent (or spatially damped) wave results in the downstream region. Here in Part III an analysis is carried out to predict the wavelength and the spatial rate of damping for this wave. The results of the analysis are verified quantitatively with seven ‘hot-start’ numerical experiments and qualitatively with seven ‘cold-start’ experiments. Weare has shown that evanescent waves occur whenever the frequency of a disturbance at a boundary exceeds the maximum frequency given by the dispersion relation. In these circumstances the ‘extended dispersion’ relation can be used to determine the rate of spatial decay.In the context of a domain consisting of two regions with different nodal spacings, the use of the group velocity concept shows that evanescent waves have no energy flux associated with them when energy is conserved.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650090706
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Internal wave reflections and transmissions arising from a non-uniform mesh. Part I: An analysis for the Crank-Nicolson linear finite element scheme (1989)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 9 (1989), S. 783-810 
    Details
    ISSN: 0271-2091
    Keywords: Non-uniform mesh ; Wave Reflection/transmission ; Crank-Nicolson finite elements ; Fourier analysis ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: This is the first of a series of three related papers dealing with some of the consequences of non-uniform meshes in a numerical model. In this paper the accuracy of the Crank-Nicolson linear finite element scheme, which is applied to the linear shallow water equations, is examined in the context of a single abrupt change in nodal spacing. The (in)accuracy is quantified in terms of reflection and transmission coefficients. An incident wave impinging on the interface between two regions with different nodal spacings is shown to give rise to no reflected waves and two transmitted waves. The analysis is verified using three different wavelengths (2Δx, 4Δx 8Δx) in three ‘hot-start’ numerical experiments with a mesh expansion factor of 2 and three experiments with a mesh contraction factor of 1/2. An energy flux analysis based on the concept of group velocity shows that energy is conserved across the interface.
    Additional Material: 16 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650090704
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Internal wave reflections and transmissions arising from a non-uniform mesh. Part II: A generalized analysis for the Crank-Nicolson linear finite element scheme (1989)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 9 (1989), S. 811-832 
    Details
    ISSN: 0271-2091
    Keywords: Fourier analysis ; Non-uniform mesh ; Wave reflection/transmission ; Group velocity ; Crank-Nicolson finite elements ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Internal wave reflections and transmissions are examined for the Crank-Nicolson linear finite element scheme applied to the linear shallow water equations in a ID domain containing an abrupt change in nodal spacing. In Part I of this series the reflection/transmission analysis was verified by some ‘hot-start’ numerical experiments. Here in Part II, however, that analysis is found wanting when it comes to providing a description of the pseudo-steady state wave configuration which develops with some ‘cold-start’ experiments. It is shown that the analysis of Part I can be extended to take in both the ‘hot-’ and ‘cold-start’ experimental results such that four essentially different wave configurations can be identified. The four configurations are discernible on the basis of group velocity. In order to be sustained, two of the configurations require one energy source whereas the other two require two energy sòurces. Numerical experiments confirmed the analysis.
    Additional Material: 14 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650090705
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Spurious numerical refraction (1995)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 21 (1995), S. 1049-1066 
    Details
    ISSN: 0271-2091
    Keywords: spurious wave refraction ; total internal reflection ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The purpose of this paper is to investigate the effect of a non-uniform mesh in two dimensions (2D). A change in mesh size will, in general, result in spurious refraction (and reflection) which is entirely numerical (rather than physical) in origin. To facilitate the analysis, the mesh geometry has been highly simplified in that only a single change in mesh size is considered. The analysis is based on a finite element wave model.The domain consists of two conterminous regions discernible only by their different nodal spacings in the x-direction. The interface between the two regions is internal to the mesh and is a straight line. The model is based upon the Crank-Nicolson linear finite element scheme applied to the second order wave equation. The results of the analysis are confirmed by numerical experiments. It is shown that under particular numerical conditions total internal reflection may occur and when this is the case, the transmitted wave is evanescent. An analysis of the energy flux associated with the incident, reflected and trasmitted waves shows that energy is conserved across the interface between the two regions.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650211104
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Analysis of spurious eigenmodes in finite element equations (1986)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 23 (1986), S. 1131-1143 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Numerical analysis of difference schemes often reveals the presence of eigenmodes which do not feature in the continuum solution. An examination of the dispersion relation shows how the spurious and physical modes interact. The behaviour of certain wave-profiles was predicted using this analysis and the results confirmed by numerical experiment.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620230610
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    Paper (German National Licenses)
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