|approved||sherwood_poster_dgconsvn.pdf||2016-04-07 13:06:24||Francois Waelbroeck|
Author: Francois L Waelbroeck
Requested Type: Poster
Submitted: 2016-02-14 11:33:16
Co-authors: C. Michoski
University of Texas at Austin
Austin , TX 78712-1
The evolution of incompressible 2D flows is the archetype and the simplest member of a class of nonlinear fluid problems that includes many multi-scale dynamical systems important in plasma physics. Such systems exhibit a variety of phenomena including turbulence, coherent structures, vortex mergers and magnetic reconnection, that have in common the importance of convective nonlinearities and the formation of fine-scale structure such as streamers and current ribbons. The approximate conservation of quantities such as magnetic helicity and potential vorticity plays an important role in the dynamics.
Discontinuous Galerkin (DG) methods provide local high-order adaptive numerical schemes for the solution of convection-diffusion problems. They combine the advantages of finite element and finite volume methods. In particular, DG methods automatically ensure the conservation of all first-order invariants provided that single-valued fluxes are prescribed at inter-element boundaries. For the 2D incompressible Euler equation, this implies that the discretized fluxes globally obey Gauss’ and Stokes’ laws exactly, and that they conserve total vorticity. Liu and Shu  have shown that combining a continuous Galerkin (CG) solution of Poisson’s equation with a central flux for the convection term leads to a stable algorithm that conserves the principal two quadratic invariants, namely the energy and enstrophy. Here, we present a discretization that applies the DG method to Poisson’s equation as well as to the vorticity equation while maintaining conservation of the quadratic invariants. Using a DG algorithm for Poisson’s equation can be advantageous when solving problems with mixed Dirichlet-Neuman boundary conditions such as for the injection of fluid through a slit (Bickley jet) or during compact toroid injection for tokamak startup.
 J.-G. Liu and C.-W. Shu, J. Comp. Phys. 160, 577-596 (2000).
Supported by US DOE Grant DE-FG02-04ER-54742.
Computer Simulation of Plasmas