Linear and nonlinear Stability analysis for finite difference discretizations of higher order Boussinesq equations

David R. Fuhrmann, Harry B. Bingham, Per A. Madsen, Per Grove Thomsen

Research output: Contribution to journalJournal articleResearchpeer-review


This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly nonlinear 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 nonlinear 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 nonlinearity 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 nonlinear analysis. The various methods of analysis combine to provide significant insight into into the numerical behavior of this rather complicated system of nonlinear PDEs.
Original languageEnglish
JournalInternational Journal for Numerical Methods in Fluids
Issue number7
Pages (from-to)751-773
Publication statusPublished - 2004


  • local nonlinear analysis
  • Boussinesq equations
  • stability analysis
  • pseudospectra
  • finite differences
  • method of lines


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