Numerical Solution of Grad-Shafranov Equation for the Distribution of Magnetic Flux in Nuclear Fusion Devices


Abstract: The magnetohydrodynamic equilibrium in an axisymmetric plasma is described by the Grad-Shafranov (GS) equation in terms of the magnetic flux. Boundary element method (BEM) is suitable for plasma equilibrium since it requires the discretization of only the plasma boundary which changes shape during the operation of an actual fusion device. In this paper, numerical solutions of the GS equation are obtained by using the boundary element method, the finite element method (FEM) and the differential quadrature method (DQM) for a rectangular plasma when the source term (current density function) on the right hand side is assumed to be a monomial. Our aim is also to find the most applicable numerical procedure between those three methods for different plasma profiles. We transform the equation to the homogeneous one with a particular solution eliminating the domain integral in the BEM formulation. For the source term containing the magnetic flux (nonlinear right hand side) an iterative procedure is made use of in the BEM and FEM formulations. It is found that the FEM gives better accuracy for a D-shape tokamak plasma whereas the BEM is more suitable for a Solov'ev tokamak plasma. The solutions agree very well with the previously published numerical solutions for a rectangular plasma.

Keywords: Grad-Shafranov equation, BEM, FEM, DQM

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