Linear and non-linear stability analysis for finite difference discretizations of high-order Boussinesq equations

Publication: Research - peer-reviewJournal article – Annual report year: 2004

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This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non-linear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann) techniques with matrix-based methods for formulations in both one and two horizontal dimensions. The matrix-based method is also extended to show the local de-stabilizing effects of the non-linear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water non-linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only moderately non-normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local non-linear analysis. The various methods of analysis combine to provide significant insight into the numerical behaviour of this rather complicated system of non-linear PDEs.
Original languageEnglish
JournalInternational Journal for Numerical Methods in Fluids
Publication date2004
Volume45
Journal number7
Pages751-773
ISSN0271-2091
StatePublished
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