Accurate calculation of the gradients of the equilibrium poloidal flux in tokamaks

This paper presents a novel method for calculating the first, second, and third derivatives of the equilibrium poloidal flux in different directions in tokamaks. The method is implemented in a new code called Equilibrium Derivative in Arbitrary Mesh (EDAM) which is designed for practical fusion applications. The spectral method is adopted along the boundary with evenly spaced angles, while unstructured triangular meshes are used inside the computational domain. A new boundary integral equation (BIE) is derived and solved numerically to obtain the first and higher-order derivatives at the boundary. Using GS equation, linear partial differential equations for the first and higher-order flux derivatives are then constructed and solved. Validation is performed using an analytical equilibrium constructed by Cicogna, which describes D-shaped plasmas with steep profiles near the boundary. The code demonstrates similar convergence rates for the first and higher-order derivatives, achieving second order accuracy. This new method has significant potential for practical fusion simulations, providing derivatives up to the third order with the required accuracy and precisely given values at any nodal points of the unstructured mesh.