Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions

Abstract

a b s t r a c tIn this work a meshless method based on the approximate particular solutions is applied to the computation of fixed boundary tokamak equilibria using Grad-Shafranov (GS) equation. The GS equation is solved for different choices of the right hand side of the equation: (i) when it is not a function of magnetic flux (i.e., Solov'ev solutions), (ii) when it is a linear function of magnetic flux, and (iii) when it is a nonlinear function of magnetic flux. For all these cases the first order derivative term in the GS equation is transferred to the right hand side such that the left hand side consists only the Laplace operator. This enables us to use the Radial Basis Functions (RBFs) in the calculation of approximate particular solutions. A linear combination of these particular solutions is taken as the solution of the GS equation and the resulting system of algebraic equations is solved iteratively because of the presence of the magnetic flux on the right hand side in all three choices. Furthermore, we use least squares approach in solving the overdetermined system of algebraic equations which alleviates the problem of ill-conditioning to a certain extent. The numerical results obtained using this method are in good agreement with the analytical solutions (where available). We find that the method is convergent, accurate and easily applicable to the irregular geometries due to its meshless character.

Plasma shape optimization for EAST tokamak using orthogonal method

“…For fixed plasma boundary and current profile, the calculation of PF current distribution essentially is a process to minimize the cost function f (𝐼 c , ψ), [14] which is defined as…”

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

“…Some other recent alternatives that have attracted attention include the hybrid approach EEC-ESC that couples Hermite elements near the plasma edge with Fourier decomposition methods in the plasma core [20], the use of meshless methods [21,22], the use of approximate particular solutions [23], and the method of fundamental solutions [24].…”