A (Dis)continuous finite element model for generalized 2D vorticity dynamics

E. Bernsen, Onno Bokhove, Jacobus J.W. van der Vegt

Research output: Book/ReportReportProfessional

49 Downloads (Pure)

Abstract

A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.
Original languageUndefined
Place of PublicationEnschede
PublisherToegepaste Wiskunde
ISBN (Print)169-2690
Publication statusPublished - 2005

Publication series

NameMemoranda
PublisherDepartment of Applied Mathematics, University of Twente
No.1787
ISSN (Print)0169-2690

Keywords

  • EWI-3607
  • METIS-227276
  • IR-65971
  • MSC-35M10
  • MSC-65M15

Cite this

Bernsen, E., Bokhove, O., & van der Vegt, J. J. W. (2005). A (Dis)continuous finite element model for generalized 2D vorticity dynamics. (Memoranda; No. 1787). Enschede: Toegepaste Wiskunde.
Bernsen, E. ; Bokhove, Onno ; van der Vegt, Jacobus J.W. / A (Dis)continuous finite element model for generalized 2D vorticity dynamics. Enschede : Toegepaste Wiskunde, 2005. (Memoranda; 1787).
@book{40092e26f661414fac00857d0da7674e,
title = "A (Dis)continuous finite element model for generalized 2D vorticity dynamics",
abstract = "A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.",
keywords = "EWI-3607, METIS-227276, IR-65971, MSC-35M10, MSC-65M15",
author = "E. Bernsen and Onno Bokhove and {van der Vegt}, {Jacobus J.W.}",
note = "Imported from MEMORANDA",
year = "2005",
language = "Undefined",
isbn = "169-2690",
series = "Memoranda",
publisher = "Toegepaste Wiskunde",
number = "1787",

}

Bernsen, E, Bokhove, O & van der Vegt, JJW 2005, A (Dis)continuous finite element model for generalized 2D vorticity dynamics. Memoranda, no. 1787, Toegepaste Wiskunde, Enschede.

A (Dis)continuous finite element model for generalized 2D vorticity dynamics. / Bernsen, E.; Bokhove, Onno; van der Vegt, Jacobus J.W.

Enschede : Toegepaste Wiskunde, 2005. (Memoranda; No. 1787).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - A (Dis)continuous finite element model for generalized 2D vorticity dynamics

AU - Bernsen, E.

AU - Bokhove, Onno

AU - van der Vegt, Jacobus J.W.

N1 - Imported from MEMORANDA

PY - 2005

Y1 - 2005

N2 - A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.

AB - A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.

KW - EWI-3607

KW - METIS-227276

KW - IR-65971

KW - MSC-35M10

KW - MSC-65M15

M3 - Report

SN - 169-2690

T3 - Memoranda

BT - A (Dis)continuous finite element model for generalized 2D vorticity dynamics

PB - Toegepaste Wiskunde

CY - Enschede

ER -

Bernsen E, Bokhove O, van der Vegt JJW. A (Dis)continuous finite element model for generalized 2D vorticity dynamics. Enschede: Toegepaste Wiskunde, 2005. (Memoranda; 1787).