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

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

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    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 languageUndefined
    Pages (from-to)502-525
    Number of pages24
    JournalJournal of computational physics
    Publication statusPublished - 2013


    • EWI-23396
    • Compatible schemes
    • Inertial waves
    • IR-86148
    • Hamiltonian structure
    • Linear Euler equations
    • METIS-297671
    • Discontinuous Galerkin method

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