Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom

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    Abstract

    The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive properties of the 1-D linear model over flat bottom and, using finite element and (pseudo-) spectral numerical codes, investigate its quality. For the optimization we restrict to the class of potentials with hyperbolic vertical profiles that are parametrized by the wavenumber. The optimal wavenumber is obtained by minimizing the kinetic energy for the given signal and produces good results for two realistic test cases. Besides this kinetic energy principle we also consider various ad-hoc least square type of minimization problems for the error of the phase or group velocity. The test cases are two examples of focussing wave groups with broad spectra for which accurate experimental data are available from MARIN hydrodynamic laboratory. To determine the quality of an 'optimized' wavenumber for the governing dynamics, we use accurate numerical simulations with the AB-equation to compare with VBM calculations for the whole range of possible wavenumbers. The comparison includes the errors in the signal at the focussing position, as well as the integrated errors of maximal and minimal wave heights along a spatial and temporal interval that is symmetric around the focussing event.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Department of Applied Mathematics
    Number of pages26
    Publication statusPublished - Feb 2010

    Publication series

    NameMemorandum / Department of Applied Mathematics
    PublisherDepartment of Applied Mathematics, University of Twente
    No.1918
    ISSN (Print)1874-4850
    ISSN (Electronic)1874-4850

    Keywords

    • Surface waves
    • Variational Boussinesq Model
    • AB-equation
    • METIS-270724
    • IR-70159
    • Optimized dispersion
    • EWI-17397

    Cite this

    Lakhturov, I., & van Groesen, E. W. C. (2010). Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom. (Memorandum / Department of Applied Mathematics; No. 1918). Enschede: University of Twente, Department of Applied Mathematics.
    Lakhturov, I. ; van Groesen, Embrecht W.C. / Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom. Enschede : University of Twente, Department of Applied Mathematics, 2010. 26 p. (Memorandum / Department of Applied Mathematics; 1918).
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    abstract = "The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive properties of the 1-D linear model over flat bottom and, using finite element and (pseudo-) spectral numerical codes, investigate its quality. For the optimization we restrict to the class of potentials with hyperbolic vertical profiles that are parametrized by the wavenumber. The optimal wavenumber is obtained by minimizing the kinetic energy for the given signal and produces good results for two realistic test cases. Besides this kinetic energy principle we also consider various ad-hoc least square type of minimization problems for the error of the phase or group velocity. The test cases are two examples of focussing wave groups with broad spectra for which accurate experimental data are available from MARIN hydrodynamic laboratory. To determine the quality of an 'optimized' wavenumber for the governing dynamics, we use accurate numerical simulations with the AB-equation to compare with VBM calculations for the whole range of possible wavenumbers. The comparison includes the errors in the signal at the focussing position, as well as the integrated errors of maximal and minimal wave heights along a spatial and temporal interval that is symmetric around the focussing event.",
    keywords = "Surface waves, Variational Boussinesq Model, AB-equation, METIS-270724, IR-70159, Optimized dispersion, EWI-17397",
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    year = "2010",
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    Lakhturov, I & van Groesen, EWC 2010, Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom. Memorandum / Department of Applied Mathematics, no. 1918, University of Twente, Department of Applied Mathematics, Enschede.

    Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom. / Lakhturov, I.; van Groesen, Embrecht W.C.

    Enschede : University of Twente, Department of Applied Mathematics, 2010. 26 p. (Memorandum / Department of Applied Mathematics; No. 1918).

    Research output: Book/ReportReportProfessional

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    N2 - The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive properties of the 1-D linear model over flat bottom and, using finite element and (pseudo-) spectral numerical codes, investigate its quality. For the optimization we restrict to the class of potentials with hyperbolic vertical profiles that are parametrized by the wavenumber. The optimal wavenumber is obtained by minimizing the kinetic energy for the given signal and produces good results for two realistic test cases. Besides this kinetic energy principle we also consider various ad-hoc least square type of minimization problems for the error of the phase or group velocity. The test cases are two examples of focussing wave groups with broad spectra for which accurate experimental data are available from MARIN hydrodynamic laboratory. To determine the quality of an 'optimized' wavenumber for the governing dynamics, we use accurate numerical simulations with the AB-equation to compare with VBM calculations for the whole range of possible wavenumbers. The comparison includes the errors in the signal at the focussing position, as well as the integrated errors of maximal and minimal wave heights along a spatial and temporal interval that is symmetric around the focussing event.

    AB - The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive properties of the 1-D linear model over flat bottom and, using finite element and (pseudo-) spectral numerical codes, investigate its quality. For the optimization we restrict to the class of potentials with hyperbolic vertical profiles that are parametrized by the wavenumber. The optimal wavenumber is obtained by minimizing the kinetic energy for the given signal and produces good results for two realistic test cases. Besides this kinetic energy principle we also consider various ad-hoc least square type of minimization problems for the error of the phase or group velocity. The test cases are two examples of focussing wave groups with broad spectra for which accurate experimental data are available from MARIN hydrodynamic laboratory. To determine the quality of an 'optimized' wavenumber for the governing dynamics, we use accurate numerical simulations with the AB-equation to compare with VBM calculations for the whole range of possible wavenumbers. The comparison includes the errors in the signal at the focussing position, as well as the integrated errors of maximal and minimal wave heights along a spatial and temporal interval that is symmetric around the focussing event.

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    KW - EWI-17397

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    Lakhturov I, van Groesen EWC. Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom. Enschede: University of Twente, Department of Applied Mathematics, 2010. 26 p. (Memorandum / Department of Applied Mathematics; 1918).