A finite volume flux coordinate independent approach

Abstract

We present a novel family of schemes as the merging between a one-dimensional advection scheme with the flux coordinate independent approach. The scheme can be used to discretize the field-aligned NavierStokes equations in three dimensions. Our approach consists of three major steps: (i) the formulation of the one-dimensional scheme in a locally field-aligned coordinate system, (ii) a numerical evaluation of the surface integrals over the field-aligned finite volumes and (iii) the introduction of smoothing into the numerical transformation operators to ensure stability of the resulting scheme. We study this approach at the example of a staggered finite volume scheme with a field-aligned cylinder as initial condition. We show superior stability and conservative properties over previous direct discretizations. In particular, the relative mass conservation is improved by several orders of magnitude. Without smoothing in the transformation operators the scheme is prone to oscillations in both parallel and perpendicular directions. In the presence of strong perpendicular gradients, additional parallel diffusion is needed to control spurious oscillations in the perpendicular planes. We provide parallel implementations for various platforms including GPUs freely in the C++ library Feltor