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

Abstract

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
Volume45
Issue number7
Pages (from-to)751-773
ISSN0271-2091
Publication statusPublished - 2004

Keywords

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

Cite this

@article{3c0c1ff670414708964eb905577a8bf9,
title = "Linear and nonlinear Stability analysis for finite difference discretizations of higher order Boussinesq equations",
abstract = "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.",
keywords = "local nonlinear analysis, Boussinesq equations, stability analysis, pseudospectra, finite differences, method of lines",
author = "Fuhrmann, {David R.} and Bingham, {Harry B.} and Madsen, {Per A.} and Thomsen, {Per Grove}",
year = "2004",
language = "English",
volume = "45",
pages = "751--773",
journal = "International Journal for Numerical Methods in Fluids",
issn = "0271-2091",
publisher = "JohnWiley & Sons Ltd.",
number = "7",

}

Linear and nonlinear Stability analysis for finite difference discretizations of higher order Boussinesq equations. / Fuhrmann, David R.; Bingham, Harry B.; Madsen, Per A.; Thomsen, Per Grove.

In: International Journal for Numerical Methods in Fluids, Vol. 45, No. 7, 2004, p. 751-773.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Linear and nonlinear Stability analysis for finite difference discretizations of higher order Boussinesq equations

AU - Fuhrmann, David R.

AU - Bingham, Harry B.

AU - Madsen, Per A.

AU - Thomsen, Per Grove

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

KW - local nonlinear analysis

KW - Boussinesq equations

KW - stability analysis

KW - pseudospectra

KW - finite differences

KW - method of lines

M3 - Journal article

VL - 45

SP - 751

EP - 773

JO - International Journal for Numerical Methods in Fluids

JF - International Journal for Numerical Methods in Fluids

SN - 0271-2091

IS - 7

ER -