Positivity preservation and advection algorithms with applications to edge plasma turbulence

The unique conditions of edge tokamak plasmas motivate the employment of efficient numerical methods that can robustly handle steep temperature and density gradients. To that end, we compare an Arakawa finite difference technique with some recent high-resolution upwind methods that are designed to minimize nonmonotonic overshoots while preserving the accuracy of solutions at smooth extrema. Versions of these algorithms are able to rigorously preserve positivity when that is physically relevant (such as for the advection of particle density or temperature). We explain in detail the use of these methods for the nonlinear Poisson bracket, an operator applicable to neutral fluid, gyrofluid, gyrokinetic, and general Hamiltonian simulations. For one-dimensional passive advection, the high-resolution upwind techniques maintain monotonicity and approach minimal levels of phase error and dissipation, especially at long wavelengths. In a two-dimensional incompressible Navier- Stokes vortex merging problem we find that extrema-preserving methods can resolve details at lower resolution than the Arakawa technique, are less dissipative than traditional finite volume methods while still minimizing overshoots and ensuring positivity, and model nonlinear cascade behavior fairly well without additional subgrid damping.