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| Titel |
Hamiltonian discontinuous Galerkin FEM for linear inertial waves |
| VerfasserIn |
S. Nurijanyan, O. Bokhove, J .J. W. van der Vegt |
| Konferenz |
EGU General Assembly 2012
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 14 (2012) |
| Datensatznummer |
250067016
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| Zusammenfassung |
| A compatible numerical scheme for a linear, three-dimensional, rotating (in)compressible
fluid flow is presented. The scheme is based on discontinuous Galerkin FEM discretisation of
corresponding Hamiltonian dynamics. The latter approach ensures a conservation of
important mathematical properties of the partial differential equations (e.g. mass, energy,
vorticity, phase space volume). Dirac theory is applied to derive the incompressible limit of
the compressible Hamiltonian system which is used as a starting point in the numerical
discretisation.
Due to the presence of Coriolis forces caused by the background rotation of the domain, the
numerical scheme admits complicated wave solutions. The waves involved are so-called
inertial waves, of relevance in oceanography and also for (filled) rotating fuel tanks. These
inertial waves display multi-scale features with chaotic attractors in zones of intense
wave activity. Thus, numerical algorithms and simulations of inertial waves are
nontrivial.
The following challenges were encountered: (i) the discretisation of an incompressible flow
or a divergence-free velocity field, a classical issue in computational fluid dynamics; (ii)
discretisation of the special, geostrophic, boundary conditions combined with no-normal flow
at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the
inertial-waves; and, (iv) the large-scale computational demands owing to the inherently
three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic
attraction.
Convergent and efficient numerical tests have been performed by comparing simulations with
the exact solutions for three-dimensional incompressible flows in rotating periodic and partly
periodic cuboids (Poincaré waves in a channel). A simulation of the linear inertial waves in a
closed rotating cuboid has been tested against semi-analytical eigenmode solutions.
Additionally, a simulation of inertial wave focusing in a hexahedron created by slanting one
wall of a cuboid will be discussed.
References:
1. S. Nurijanyan, J. J. W. van der Vegt, O. Bokhove: Hamiltonian discontinuous
Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves.
http://eprints.eemcs.utwente.nl/21124/ (submitted to J. Comp. Phys.)
2. Y. Xu, J. J. W. van der Vegt, O. Bokhove, Discontinuous Hamiltonian Finite Element
Method for linear hyperbolic systems, J. Sci. Comput. 35 (2008) 241–265.
3. L. Pesch, A. Bell, W. E. H. Solie, V. R. Ambati, O. Bokhove and J. J. W. van der Vegt,
hpGEM —A software framework for discontinuous Galerkin finite element Methods, ACM
Transactions on Mathematical Software, 33(4) (2007).
4. O. Bokhove, On balanced models in Geophysical Fluid Dynamics: Hamiltonian
formulation, constraints and formal stability, Large-Scale Atmosphere-Ocean Dynamics: Vol
II: Geometric Methods and Models. Ed.J. Norbury and I. Roulstone, Cambridge University
Press, Cambridge, (2002) 1–63. |
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