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 verications concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls.
|Name||Memorandum / Department of Applied Mathematics|
|Publisher||University of Twente, Department of Applied Mathematics|
- Hamiltonian structure
- Discontinuous Galerkin method
- Compatible schemes
- Linear Euler equations
- Inertial waves