Dirichlet boundary conditions for arbitrary-shaped boundaries in stellarator-like magnetic fields for the Flux-Coordinate Independent method

TL;DR

This paper introduces a novel method for applying Dirichlet boundary conditions in the Flux-Coordinate Independent (FCI) approach, enabling accurate simulations in complex 3D magnetic geometries like stellarators without singularities.

Contribution

The paper presents the Leg Value Fill (LVF) boundary scheme integrated into FCI, verified with the Method of Manufactured Solutions, and demonstrates its effectiveness in non-axisymmetric stellarator-like magnetic fields.

Findings

LVF scheme accurately enforces boundary conditions in arbitrary geometries.

The FCI method with LVF reproduces expected diffusion scaling in stellarator configurations.

Implementation in BOUT++ is validated with error analysis and flux surface tracing.

Abstract

We present a technique for handling Dirichlet boundary conditions with the Flux Coordinate Independent (FCI) parallel derivative operator with arbitrary-shaped material geometry in general 3D magnetic fields. The FCI method constructs a finite difference scheme for $\nabla_\Vert$ by following field lines between planes and interpolating within planes. Doing so removes the need for field-aligned coordinate systems that suffer from singularities in the metric tensor at null points in the magnetic field (or equivalently, when $q \to \infty$). One cost of this method is that as the field lines are not on the mesh, they may leave the domain at any point between neighbouring planes, complicating the application of boundary conditions. The Leg Value Fill (LVF) boundary condition scheme presented here involves an extrapolation/interpolation of the boundary value onto the field line end point. The usual finite difference scheme can then be used unmodified. We implement the LVF scheme in BOUT++ and use the Method of Manufactured Solutions to verify the implementation in a rectangular domain, and show that the error scaling of the finite difference scheme isn't modified. The use of LVF for arbitrary wall geometry is outlined. We also demonstrate the feasibility of using the FCI approach in non-axisymmetric configurations for a simple diffusion model in a "straight stellarator" magnetic field. A Gaussian blob diffuses along the field lines, tracing out flux surfaces. Dirichlet boundary conditions impose a last closed flux surface (LCFS) that confines the density. Including a poloidal limiter moves the LCFS to a smaller radius. The expected scaling of the numerical perpendicular diffusion, which is a consequence of the FCI method, in stellarator-like geometry is recovered. A novel technique for increasing the parallel resolution during post-processing is also described.