Chebyshev spectral collocation methods for nonlinear isothermal magnetostatic atmospheres

The equations of magnetohydrodynamic (MHD) equilibria for a plasma in gravitational field are investigated. For equilibria with one ignorable spatial coordinate, the MHD equations are reduced to a single nonlinear elliptic equation (NLPDE) for the magnetic flux u, known as the Grad-Shafranov equation. The Chebyshev spectral collocation methods (CSCMs) are described and applied to obtain numerical solutions for nonlinear boundary value problems modelling two classes of isothermal magnetostatic atmospheres, in which the current density J is proportional to the exponential of the magnetic flux and moreover falls off exponentially with distance vertical to the base, with an “e-folding” distance equal to the gravitational scale height, for the first class and proportional to the sinh(u) for the second class. The accuracy and efficiency of this Chebyshev approach are compared favorably with those of the standard finite-difference methods.