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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:241

Article

Nurijanyan, S.¹  van der Vegt, J. J. W.¹  Bokhove, O.^1,2 

[1] Univ Twente, Dept Appl Math, NL-7500 AE Enschede, Netherlands

[2] Univ Leeds, Sch Math, Leeds LS2 9JT, W Yorkshire, England

关键词: Linear Euler equations;    Hamiltonian structure;    Discontinuous Galerkin method;    Inertial waves;    Compatible schemes;   

DOI  :  10.1016/j.jcp.2013.01.017

来源: Elsevier

【 摘 要 】

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 (Poincare 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. (C) 2013 Elsevier Inc. All rights reserved.

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