A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad–Shafranov equation by extension from polygonal subdomains

“…Below we describe algorithmically the simplest form of the acceleration-which is the version implemented in our solver-but we refer the reader to the works by Kelly and Toth [26], and Walker and Ni [27], where the method is studied in detail. If we denote by a prescribed tolerance, by ψ 0 the initial input, by (−∆ * ) −1 the solution operator to (15) described above, and by ψ h the final, approximate solution to the non-linear problem then, in its simplest form, the acceleration algorithm that uses m previous iterates can be described as follows:…”

“…Before describing the adaptive algorithm, we will discuss the problem on a fixed, uniform and embedded polygonal mesh. Most of the details have been described in [15]. However, starting with this standard case will allow us to introduce the notation and the fundamental ideas underpinning our HDG approach, as well as our treatment of curved boundaries and the iterative method to treat the non-linearity of the equation.…”

“…In this article, we present the first numerical Grad-Shafranov solver which satisfies these four performance requirements. To achieve this goal, we added adaptive refinement capabilities to the Hybridizable Discontinuous Galerkin (HDG) Grad-Shafranov solver we originally presented in [15]. The solver relies on reformulating the problem as a first order system [18], which takes the form…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…More recently, there has been growing interest towards the analysis and simulation of quasilinear and semilinear problems, including the quasilinear p-Laplace operator [95,216] and the semilinear Grad-Shafranov equation [233,234]. To reduce the computational cost of semilinear problems, the interpolatory HDG method was recently devised introducing an interpolation procedure for the efficient and accurate approximation of nonlinear terms [54,100].…”

“…Computational physics community is showing increasing interest towards the application of hybrid discretisation methods to the simulation of magnetic plasma physics. Promising preliminary results concerning the HDG approximation of the Grad-Shafranov equation in axisymmetric confinement devices modelling fusion reactors are described in [233,234]. In the context of magnetohydrodynamics (MHD), an HDG method for steady-state linearised incompressible MHD equations is proposed in [180].…”