Exponential reconstructions and positivity for discontinuous Galerkin algorithms, and linear benchmarks of the Gkeyll gyrokinetic code*

One of the challenges of simulating turbulence in the edge regions of tokamaks is handling the large amplitude fluctuations that can occur and avoiding negative density overshoots, which might cause various problems.   One of the attractive features of discontinuous Galerkin (DG) algorithms is that they can conserve particles and energy exactly for Hamiltonian systems even if limiters are used for fluxes at cell boundaries.  To preserve the realizability of the solution within a cell, there are situations where it is actually necessary to enhance the boundary flux, not limit it.  We show a new method that accomplishes this with exponential reconstructions, while still preserving exact particle and energy conservation.  Exponential reconstructions are also helpful for handling the infinite velocity domain.  We demonstrate the algorithm in 1D with a series of tests, show that this gives a systematic local method of ensuring positivity and realizability of the solution, and show some tests in multiple dimensions.  Finally, we demonstrate some linear benchmark tests of the full 3D+2v Gkeyll gyrokinetic code, comparing with the toroidal ITG linear dispersion relation over a range of parameters.