Grad-Shafranov equation: MHD simulation of the new solution obtained from the Fadeev and Naval models

This article aims to obtain a new analytical solution of a specific form of the Grad-Shafranov (GS) equation using Walker's formula. The new solution has magnetic field lines with X-type neutral points, magnetic islands and singular points. The singular points are located on the x-axis. The X-points and the center of the magnetic islands do not appear on the x-axis an island appears at $z>0$ and the other two at $z<0$.  The aforementioned property allows us to use this solution as an initial condition at $t=0$ s in an magnetohydrodynamic (MHD) numerical simulation by excluding the singular points of the solution, i.e., the x-axis, and maintaining the magnetic structure of the islands, as well as the X-type neutral points.  For this, we numerically solve the equations of the classical ideal MHD in two dimensions using the Newtonian CAFE code.  The code is based on high resolution shock capturing methods using the Harten-Lax-van Leer-Einfeldt (HLLE) flux formula combined with MINMOD...