Fast BIE method for computing force-free magnetic fields in toroidal geometries

We develop a boundary integral equation solver for computing Taylor relaxed states in a given region bounded by flux surfaces. The region can be bounded by a single toroidal surface or it can be the region between two nested toroidal surfaces. Such solutions can be used to study ideal magneto-hydrodynamic (MHD) equilibria with stepped pressure profiles in magnetically confined plasmas in tokamaks and stellarators.

In our scheme, we compute Taylor states using the generalized Debye source formulation for the time-harmonic Maxwell's equations, which results in a well-conditioned second-kind boundary integral equations. The boundary integral formulation requires fewer unknowns since we do not need to discretize the entire volume and this results in significant savings in work. However, the computation of the boundary integral operator requires efficient high-order quadratures for singular and near singular integrals on complex three-dimensional surfaces. We will discuss fast algorithms for such quadratures. We will present numerical results to show the accuracy, efficiency and scalability of our method for several challenging geometries.