Toroidal figures of equilibrium from a second-order accurate, accelerated SCF method with subgrid approach

We compute the structure of a self-gravitating torus with polytropic equation-of-state (EOS) rotating in an imposed centrifugal potential. The Poisson-solver is based on isotropic multigrid with optimal covering factor (fluid section-to-grid area ratio). We work at $2$nd-order in the grid resolution for both finite difference and quadrature schemes. For soft EOS (i.e. polytropic index $n \ge 1$), the underlying $2$nd-order is naturally recovered for Boundary Values (BVs) and any other integrated quantity sensitive to the mass density (mass, angular momentum, volume, Virial Parameter, etc.), i.e. errors vary with the number $N$ of nodes per direction as $\sim 1/N^2$. This is, however, not observed for purely geometrical quantities (surface area, meridional section area, volume), unless a subgrid approach is considered (i.e. boundary detection). Equilibrium sequences are also much better described, especially close to critical rotation. Yet another technical effort is required for hard EOS ($n < 1$), due to infinite mass density gradients at the fluid surface. We fix the problem by using kernel splitting. Finally, we propose an accelerated version of the SCF-algorithm based on a node-by-node pre-conditionning of the mass density at each step. The computing time is reduced by a factor $2$ typically, regardless of the polytropic index. There is a priori no obstacle to applying these results and techniques to ellipsoidal configurations and even to $3$D-configurations.