Solving the Grad–Shafranov equation with spectral elements

The development of a generalized two dimensional MHD equilibrium solver within the nimrod framework [Sovinec, et al., J. Comput. Phys. 195 (2004) 355] is discussed. Spectral elements are used to represent the poloidal plane. To permit the generation of spheromak and other compact equilibria, special consideration is given to ensure regularity at the geometric axis (R=0)(R=0). The scalar field Λ=ψ/R2Λ=ψ/R2 is used as the dependent variable to express the Grad–Shafranov operator as a total divergence. With the correct gauge, regularity along the geometric axis is satisfied. The convergence properties of the spectral elements are investigated by comparing numerically generated equilibria against known analytic solutions. Equilibria accurate to double precision error are generated with sufficient resolution. Depending on the equilibrium, either geometric or algebraic convergence is observed as the polynomial degree of the spectral-element basis is increased.