Discontinuous Galerkin method for linear free-surface gravity waves

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Abstract

In this paper, we discuss a discontinuous Galerkin finite element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the $L^2$-norm is in both cases of optimal order and proportional to $O(\triangle t^2+h^{p+1})$, without the need for a separate velocity reconstruction, with $p$ the polynomial order, $h$ the mesh size and $\triangle t$ the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages44
ISBN (Print)0169-2690
Publication statusPublished - 2004

Publication series

NameMemoranda
PublisherDepartment of Applied Mathematics, University of Twente
No.1721
ISSN (Print)0169-2690

Keywords

  • MSC-76M10
  • MSC-76B15
  • MSC-76B07
  • EWI-3541
  • IR-65906
  • MSC-65N30
  • METIS-218343
  • MSC-65N15
  • MSC-65N12
  • MSC-35J05

Cite this

van der Vegt, J. J. W., & Tomar, S. K. (2004). Discontinuous Galerkin method for linear free-surface gravity waves. (Memoranda; No. 1721). Enschede: University of Twente, Department of Applied Mathematics.
van der Vegt, Jacobus J.W. ; Tomar, S.K. / Discontinuous Galerkin method for linear free-surface gravity waves. Enschede : University of Twente, Department of Applied Mathematics, 2004. 44 p. (Memoranda; 1721).
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title = "Discontinuous Galerkin method for linear free-surface gravity waves",
abstract = "In this paper, we discuss a discontinuous Galerkin finite element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the $L^2$-norm is in both cases of optimal order and proportional to $O(\triangle t^2+h^{p+1})$, without the need for a separate velocity reconstruction, with $p$ the polynomial order, $h$ the mesh size and $\triangle t$ the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.",
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author = "{van der Vegt}, {Jacobus J.W.} and S.K. Tomar",
note = "Imported from MEMORANDA",
year = "2004",
language = "Undefined",
isbn = "0169-2690",
series = "Memoranda",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1721",

}

van der Vegt, JJW & Tomar, SK 2004, Discontinuous Galerkin method for linear free-surface gravity waves. Memoranda, no. 1721, University of Twente, Department of Applied Mathematics, Enschede.

Discontinuous Galerkin method for linear free-surface gravity waves. / van der Vegt, Jacobus J.W.; Tomar, S.K.

Enschede : University of Twente, Department of Applied Mathematics, 2004. 44 p. (Memoranda; No. 1721).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - Discontinuous Galerkin method for linear free-surface gravity waves

AU - van der Vegt, Jacobus J.W.

AU - Tomar, S.K.

N1 - Imported from MEMORANDA

PY - 2004

Y1 - 2004

N2 - In this paper, we discuss a discontinuous Galerkin finite element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the $L^2$-norm is in both cases of optimal order and proportional to $O(\triangle t^2+h^{p+1})$, without the need for a separate velocity reconstruction, with $p$ the polynomial order, $h$ the mesh size and $\triangle t$ the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.

AB - In this paper, we discuss a discontinuous Galerkin finite element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the $L^2$-norm is in both cases of optimal order and proportional to $O(\triangle t^2+h^{p+1})$, without the need for a separate velocity reconstruction, with $p$ the polynomial order, $h$ the mesh size and $\triangle t$ the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.

KW - MSC-76M10

KW - MSC-76B15

KW - MSC-76B07

KW - EWI-3541

KW - IR-65906

KW - MSC-65N30

KW - METIS-218343

KW - MSC-65N15

KW - MSC-65N12

KW - MSC-35J05

M3 - Report

SN - 0169-2690

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BT - Discontinuous Galerkin method for linear free-surface gravity waves

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

van der Vegt JJW, Tomar SK. Discontinuous Galerkin method for linear free-surface gravity waves. Enschede: University of Twente, Department of Applied Mathematics, 2004. 44 p. (Memoranda; 1721).