Premium
Linear and non‐linear stability analysis for finite difference discretizations of high‐order Boussinesq equations
Author(s) -
Fuhrman David R.,
Bingham Harry B.,
Madsen Per A.,
Thomsen Per G.
Publication year - 2004
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.713
Subject(s) - von neumann stability analysis , mathematics , conservative vector field , eigenvalues and eigenvectors , linear stability , linear system , computation , mathematical analysis , numerical analysis , matrix (chemical analysis) , stability (learning theory) , numerical stability , nonlinear system , finite difference , equivalence (formal languages) , compressibility , mechanics , physics , computer science , algorithm , materials science , quantum mechanics , machine learning , discrete mathematics , composite material
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. Copyright © 2004 John Wiley & Sons, Ltd.