Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves

S. Nurijanyan, Jacobus J.W. van der Vegt, Onno Bokhove

Research output: Book/ReportReport

Abstract

A discontinuous Galerkin ﬿nite element method (DGFEM) has been developed and tested for linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions. The numerical challenges concern: (i) discretisation of a divergence-free velocity ﬿eld; (ii) discretisation of geostrophic boundary conditions combined with no-normal flow at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the inertial-waves; and, (iv) large-scale computational demands owing to the three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic attraction. These issues have been resolved: (i) by employing Dirac’s method of constrained Hamiltonian dynamics to our DGFEM for linear, compressible flows, thus enforcing the incompressibility constraints; (ii) by enforcing no-normal flow at solid walls in a weak form and geostrophic tangential flow —along the wall; (iii) by applying a symplectic time discretisation; and, (iv) by combining PETSc’s linear algebra routines with our high-level software. We compared our simulations with exact solutions of three-dimensional compressible and incompressible flows, in (non)rotating periodic and partly periodic cuboids (Poincar´e waves). Additional veri﬿cations concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls.
LanguageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages48
StatePublished - Dec 2011

Publication series

NameMemorandum / Department of Applied Mathematics
PublisherUniversity of Twente, Department of Applied Mathematics
No.1970
ISSN (Print)1874-4850
ISSN (Electronic)1874-4850

Keywords

  • IR-79126
  • METIS-284943
  • EWI-21124
  • Hamiltonian structure
  • Discontinuous Galerkin method
  • Compatible schemes
  • Linear Euler equations
  • Inertial waves

Cite this

Nurijanyan, S., van der Vegt, J. J. W., & Bokhove, O. (2011). Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves. (Memorandum / Department of Applied Mathematics; No. 1970). Enschede: University of Twente, Department of Applied Mathematics.
Nurijanyan, S. ; van der Vegt, Jacobus J.W. ; Bokhove, Onno. / Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves. Enschede : University of Twente, Department of Applied Mathematics, 2011. 48 p. (Memorandum / Department of Applied Mathematics; 1970).
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Nurijanyan, S, van der Vegt, JJW & Bokhove, O 2011, Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves. Memorandum / Department of Applied Mathematics, no. 1970, University of Twente, Department of Applied Mathematics, Enschede.

Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves. / Nurijanyan, S.; van der Vegt, Jacobus J.W.; Bokhove, Onno.

Enschede : University of Twente, Department of Applied Mathematics, 2011. 48 p. (Memorandum / Department of Applied Mathematics; No. 1970).

Research output: Book/ReportReport

TY - BOOK

T1 - Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves

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N2 - A discontinuous Galerkin ﬿nite element method (DGFEM) has been developed and tested for linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions. The numerical challenges concern: (i) discretisation of a divergence-free velocity ﬿eld; (ii) discretisation of geostrophic boundary conditions combined with no-normal flow at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the inertial-waves; and, (iv) large-scale computational demands owing to the three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic attraction. These issues have been resolved: (i) by employing Dirac’s method of constrained Hamiltonian dynamics to our DGFEM for linear, compressible flows, thus enforcing the incompressibility constraints; (ii) by enforcing no-normal flow at solid walls in a weak form and geostrophic tangential flow —along the wall; (iii) by applying a symplectic time discretisation; and, (iv) by combining PETSc’s linear algebra routines with our high-level software. We compared our simulations with exact solutions of three-dimensional compressible and incompressible flows, in (non)rotating periodic and partly periodic cuboids (Poincar´e waves). Additional veri﬿cations concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls.

AB - A discontinuous Galerkin ﬿nite element method (DGFEM) has been developed and tested for linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions. The numerical challenges concern: (i) discretisation of a divergence-free velocity ﬿eld; (ii) discretisation of geostrophic boundary conditions combined with no-normal flow at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the inertial-waves; and, (iv) large-scale computational demands owing to the three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic attraction. These issues have been resolved: (i) by employing Dirac’s method of constrained Hamiltonian dynamics to our DGFEM for linear, compressible flows, thus enforcing the incompressibility constraints; (ii) by enforcing no-normal flow at solid walls in a weak form and geostrophic tangential flow —along the wall; (iii) by applying a symplectic time discretisation; and, (iv) by combining PETSc’s linear algebra routines with our high-level software. We compared our simulations with exact solutions of three-dimensional compressible and incompressible flows, in (non)rotating periodic and partly periodic cuboids (Poincar´e waves). Additional veri﬿cations concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls.

KW - IR-79126

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KW - Linear Euler equations

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BT - Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves

PB - University of Twente, Department of Applied Mathematics

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ER -

Nurijanyan S, van der Vegt JJW, Bokhove O. Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves. Enschede: University of Twente, Department of Applied Mathematics, 2011. 48 p. (Memorandum / Department of Applied Mathematics; 1970).