2016 CMS Winter Meeting
Niagara Falls, December 2 - 5, 2016
The resulting variational integrators allow for a discrete version of Kelvin circulation theorem, are applicable to irregular meshes and exhibit excellent long term energy behavior. We then report a series of tests for these models on regular and irregular meshes with strongly deformed cells. In the benchmark test of hydrostatic adjustment we verify that the models correctly represent the dispersion relations. We further show that these schemes correctly capture the Kelvin Helmholtz instability. Finally, on the rising and falling bubble test cases we illustrate the models' accurate representation of advection dominated processes. For these test cases and grid types, these variational models show excellent long time energy conservation.
This is a joint work with Fran\c{c}ois Gay-Balmaz (Laboratoire de M\'et\'eorologie Dynamique, \'Ecole Normale Sup\'erieure/CNRS, Paris, France).
For the first, we will introduce geometric degrees of freedom, which are associated to geometric objects (points, lines, surfaces and volumes), and then establish their relation to differential forms. It will be shown that we can construct discrete polynomial function spaces of arbitrary degree associated to these geometric degrees of freedom. These function spaces constitute a discrete de Rham complex:
\[ \mathbb{R}\longrightarrow V^{0}_{h}\subseteq H(\nabla,\Omega) \overset{\nabla}{\longrightarrow} V^{1}_{h}\subseteq H(\nabla\times,\Omega) \overset{\nabla\times}{\longrightarrow} V^{2}_{h}\subseteq H(\nabla\dot,\Omega)\overset{\nabla\cdot}{\longrightarrow} V^{3}_{h}\subseteq L^{2}(\Omega)\rightarrow 0\,. \]
In this way it is possible to exactly discretize topological equations even on highly deformed meshes. All approximation errors are included in the constitutive equations, which are encoded in the Hodge-$\star$ operator. This leads to discretizations that exactly preserve the divergence free constraint of velocity fields in incompressible flow problems and of magnetic fields in electromagnetic problems, for example.
For the second, the Navier-Stokes equations will be used as an example and we will show that although at the continuous level all equivalent formulations are equally good, at the discrete level, the choice of a particular formulation has a fundamental impact on the conservation properties of the discretization.
These ideas will be applied to the solution of: e.g. Poisson equation, Maxwell eigenvalue problem, Darcy flow, fusion plasma equilibrium and Navier-Stokes equations.
(Joint work with Robert McLachlan, Massey University, New Zealand.)
This is joint work with A. Bihlo (Memorial University at Newfoundland) and J.-C. Nave (McGill University).