Fixed boundary Grad-Shafranov solver using finite difference method in nonhomogeneous meshgrid

Abstract

In this work we present a numerical scheme to solve the Grad-Shafranov equation which correspond to magnetohydrodynamic equilibrium equation for a two-dimensional plasma. A typical case are the toroidal plasma in magnetic confinement devices used in thermonuclear fusion well known as Tokamaks. The proposed numerical scheme is based on the finite-difference method in nonhomogeneous meshgrid, which is adjusted to the fixed plasma boundary with "D-shape". The solution of the Grad-Shafranov equation is obtained using the successive over-relaxation method, usually applied to solve Poisson equation's problems. The values of the total plasma current and pressure in the magnetic axis are conserved in each iteration of the convergence process. The scheme is validated by direct comparison with the analytical result obtained by Soloviev.