Abstract
A discontinuous Galerkin finite element method (DGFEM) has been developed and tested for the linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions, which poses numerical challenges.
These challenges concern: (i) discretisation of a divergence-free velocity field; (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, for example: (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; and, (iii) by applying a symplectic time discretisation.
We compared our simulations with exact solutions of three-dimensional incompressible flows, in (non) rotating periodic and partly periodic cuboids (Poincaré waves). Additional verifications concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls. Finally, a simulation in a tilted rotating tank, yielding more complicated wave dynamics, demonstrates the potential of our new method.
Original language | Undefined |
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Pages (from-to) | 502-525 |
Number of pages | 24 |
Journal | Journal of computational physics |
Volume | 241 |
DOIs | |
Publication status | Published - 2013 |
Keywords
- EWI-23396
- Compatible schemes
- Inertial waves
- IR-86148
- Hamiltonian structure
- Linear Euler equations
- METIS-297671
- Discontinuous Galerkin method